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✓ 31 results · all verified · 25 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Scalar Conservation Laws and Entropy Solutions

1 · Prerequisites

2 · Summary

This page develops the theory of scalar conservation laws $u_{t} + {div}_{x} f (u) = 0$ from the definitions through existence, uniqueness and wave structure, with the vanishing-viscosity method as its backbone. It fixes the flux and the distributional weak formulation, verifies that classical solutions are weak solutions, derives the constancy of $u$ along characteristics together with the gradient-catastrophe formula for one-dimensional solutions, and sets up piecewise $C^{1}$ shocks with one-sided traces and the space--time Rankine--Hugoniot condition. Non-uniqueness of the weak formulation is demonstrated before any selection principle is introduced. Convex entropy--entropy-flux pairs and the viscous entropy-dissipation identity lead to the Kruzhkov notion: bounded solutions satisfying the entropy inequalities for $η_{k} (s) = ∣ s - k ∣$ , with the strong local $L^{1}$ initial trace.

The constructive half of the page proves the global classical solvability of the viscous Cauchy problem with smooth data, its uniform $L^{\infty}$ , mass and energy bounds, and the uniform contraction of spatial translates; the doubling-variables (Kato) inequality for two entropy solutions yields the local $L^{1}$ contraction, from which uniqueness, comparison, order preservation and finite propagation follow. Uniform bounds and the translate and time moduli give local precompactness of vanishing-viscosity families, the global $L^{1}$ contraction, and finally the existence of a bounded Kruzhkov entropy solution for $L^{1} \cap L^{\infty}$ data by passing the weak equation and the viscous entropy balance to the limit. The structure theory then characterises admissible jumps by the flux-chord inequality, solves the Riemann problem for strictly convex fluxes with its shock and centred-rarefaction profiles, proves Oleinik's one-sided estimate $\partial_{x} u \leq (κ t)^{- 1}$ as an equivalent entropy condition under uniform convexity on the state range, derives the Lax shock inequalities, and establishes the one-dimensional Hamilton--Jacobi correspondence between entropy solutions and primitives of viscosity solutions.

The closing items record quantitative consequences: mass conservation for compactly supported integrable data, invariance of the entropy inequality under additive constants in the entropy flux, the $L^{\infty}$ maximum bound, the entropy solution semigroup on $L^{1} \cap L^{\infty}$ with its extension to $L^{1}$ for globally Lipschitz flux with $f (0) = 0$ , and strong $L^{1}$ continuity of the orbits. Countable Choice and Dependent Choice are declared where the analytic interfaces require them, including inheritance by consumers. The Hamilton--Jacobi correspondence is proved through viscous primitives and localized Hopf--Lax formulas; no viscosity/entropy equivalence is silently imported.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)Open item page →

Scalar conservation laws, fluxes and Cauchy data

Definition

Let $n \geq 1$ and let $Π = O \times (a, b) \subseteq R^{n} \times R$ be an open space--time cylinder. The equation $u_{t} + {div}_{x} f (u) = 0 in Π$ is the scalar conservation law in conservation form, where the unknown $u ⁣ : Π \to R$ is the conserved quantity and the flux is a map $f ⁣ : R \to R^{n}$ . The classical expression is used when $u$ and $f \circ u$ are $C^{1}$ ; the distributional expression is used whenever $u, f (u) \in L_{l o c}^{1} (Π)$ . In particular, if $u$ is bounded measurable and $f$ is continuous, then $f (u)$ is locally integrable and its distributional divergence is defined (A locally integrable function on $R^{n}$ , The space $L^{p} (μ)$ as the quotient by null functions).

A Cauchy problem is posed separately on $Π_{T} = R^{n} \times (0, T)$ and has initial datum $u_{0} \in L^{\infty} (R^{n}) \cap L_{l o c}^{1} (R^{n})$ , understood as an equivalence class; the weak formulation records it in the initial boundary term and the entropy formulation uses a strong local $L^{1}$ trace. If $f \in C^{1} (R; R^{n})$ and $u \in C^{1}$ , the chain rule gives the equivalent quasilinear equation $u_{t} + f^{'} (u) \cdot \nabla u = 0$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Divergence and curl of a $C^{1}$ vector field). This equivalence is not asserted for discontinuous $u$ : the product of $f^{'} (u)$ with a distributional gradient is not generally defined, while ${div}_{x} f (u)$ is defined distributionally whenever $f (u) \in L_{l o c}^{1}$ .

The one-dimensional case is $u_{t} + f (u)_{x} = 0$ with scalar flux $f ⁣ : R \to R$ . Whenever the Riemann theory, the Rankine--Hugoniot condition or the characteristic formula below is invoked, $f$ is assumed at least $C^{1}$ (and $C^{2}$ where $f^{'}$ or $f^{''}$ is differentiated); adding a constant vector to $f$ does not change the equation because its divergence is zero.

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)Open item page →

Distributional weak solutions of the Cauchy problem

Definition

Let $n \geq 1$ , $T > 0$ , $Π_{T} = R^{n} \times (0, T)$ , let $f ⁣ : R \to R^{n}$ be continuous and let $u_{0} \in L^{\infty} (R^{n}) \cap L_{l o c}^{1} (R^{n})$ . A bounded measurable $u ⁣ : Π_{T} \to R$ is a distributional weak solution of the Cauchy problem $u_{t} + {div}_{x} f (u) = 0$ , $u (\cdot, 0) = u_{0}$ , if for every $φ \in C_{c}^{\infty} (R^{n} \times (- \infty, T))$ --- test functions whose support may meet the initial plane $t = 0$ --- one has $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla_{x} φ) d x d t + \int_{R^{n}} u_{0} (x) φ (x, 0) d x = 0.$ Equivalently $u_{t} + {div}_{x} f (u) = 0$ in $D^{'} (Π_{T})$ , together with the displayed initial boundary term (Distribution, Distributional derivative, Test function space d of an open set). The integrals are absolutely convergent: $u$ is bounded, $f (u)$ is bounded on the bounded range of $u$ , and $φ$ has compact support, so the pairings are $L_{l o c}^{1}$ pairings and are representative-independent (A locally integrable function on $R^{n}$ , The space $L^{p} (μ)$ as the quotient by null functions; the product-space identities are those of Fubini's theorem for L^1 functions on a sigma-finite product and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product). The condition is an equality in the $L_{l o c}^{1}$ classes of $u$ and $f (u)$ ; it involves no pointwise assignment of $u$ on ${t = 0}$ , and the initial datum enters only through the boundary term (Scalar conservation laws, fluxes and Cauchy data). For the entropy formulation of this page the initial condition is instead imposed as a strong local $L^{1}$ trace (Kruzhkov entropy solutions).

PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Classical solutions are distributional weak solutions, and conversely

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the injectivity interface below. Let $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ and $u_{0} \in L^{\infty} (R^{n}) \cap L_{l o c}^{1} (R^{n})$ .

(i) If $u \in C^{1} (Π_{T}) \cap C^{0} ([0, T]; L_{l o c}^{1} (R^{n})) \cap L^{\infty} (Π_{T})$ satisfies $u_{t} + {div}_{x} f (u) = 0$ pointwise in $Π_{T}$ and has initial trace $u (\cdot, t) \to u_{0}$ in $L_{l o c}^{1}$ as $t ↓ 0$ , then $u$ is a distributional weak solution in the sense of Distributional weak solutions of the Cauchy problem.

(ii) Conversely, if $u$ is a bounded distributional weak solution, $u \in C^{1} (Π_{T})$ , and $u (\cdot, t)$ has an $L_{l o c}^{1}$ -continuous trace ${\overset{ˉ}{u}}_{0}$ as $t ↓ 0$ , then $u_{t} + {div}_{x} f (u) = 0$ pointwise in $Π_{T}$ and ${\overset{ˉ}{u}}_{0} = u_{0}$ as $L_{l o c}^{1}$ equivalence classes.

The initial-trace conclusion is an almost-everywhere class equality; no pointwise representative on $t = 0$ is asserted (Scalar conservation laws, fluxes and Cauchy data).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ , $u_{0} \in L^{\infty} \cap L_{l o c}^{1}$ , a classical solution $u \in C^{1} (Π_{T})$ satisfying $u_{t} + {div}_{x} f (u) = 0$ pointwise with an $L_{l o c}^{1}$ initial trace (i), and, in (ii), a bounded distributional weak solution $u \in C^{1} (Π_{T})$ with an $L_{l o c}^{1}$ -continuous initial trace ${\overset{ˉ}{u}}_{0}$ .

[F1]

For a $C^{1}$ function $u$ the composition $f (u)$ is $C^{1}$ with ${div}_{x} f (u) = f^{'} (u) \cdot \nabla_{x} u$ , by the chain rule and the algebra of derivatives; the Laplacian-free flux is locally integrable on compact space--time boxes (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Scalar conservation laws, fluxes and Cauchy data).

[F2]

Integration by parts on a box follows coordinatewise from The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ and Fubini's theorem for L^1 functions on a sigma-finite product applied to the $C^{1}$ products $u φ$ and $f_{i} (u) φ$ . Compact support kills spatial and terminal faces. No surface divergence theorem is used.

[F3]

A continuous function on an open set whose integral against every compactly supported smooth test function vanishes is identically zero there; testing against a translate of a fixed nonzero smooth compactly supported bump gives a nonzero pairing where the function does not vanish (Explicit compactly supported smooth cutoffs).

[F4]

Under Countable Choice, two locally integrable functions with equal pairings against every smooth compactly supported test represent the same almost-everywhere class (Locally integrable functions embed in distributions).

Proof

technique · direct

1.1F1F2

Fix $φ \in C_{c}^{\infty} (R^{n} \times (- \infty, T))$ and choose $Q_{R} = (- R, R)^{n}$ and $τ$ with $supp φ \subset Q_{R} \times (- \infty, τ)$ , $τ < T$ . Multiplying the pointwise equation by $φ$ and integrating over $Q_{R} \times (δ, τ)$ , $0 < δ < τ$ , [F2] gives $0 = \int_{Q_{R}} u φ ∣_{δ}^{τ} d x - \int_{δ}^{τ} ⁣ ⁣ \int_{Q_{R}} u φ_{t} - \int_{δ}^{τ} ⁣ ⁣ \int_{Q_{R}} f (u) \cdot \nabla_{x} φ$ , since $φ$ vanishes on the lateral and terminal faces.

1.2F1F3given

For (ii), test the weak identity with $φ \in C_{c}^{\infty} (Π_{T})$ (so $φ = 0$ near $t = 0$ ): integration by parts over the support of $φ$ gives $0 = \int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla_{x} φ) = - ⟨ u_{t} + {div}_{x} f (u), φ ⟩$ , where the residual $R : = u_{t} + {div}_{x} f (u)$ is continuous by [F1]. By [F3] applied on the open set $Π_{T}$ , $R \equiv 0$ , so the equation holds pointwise.

2.1F2givenstep 1.1

In step 1.1 the terminal term vanishes and $φ (\cdot, δ) \to φ (\cdot, 0)$ uniformly on the compact spatial support while $u (\cdot, δ) \to u_{0}$ in $L_{l o c}^{1}$ ; letting $δ ↓ 0$ gives $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla_{x} φ) d x d t + \int_{R^{n}} u_{0} φ (\cdot, 0) d x = 0$ , which is the weak formulation for this test function. As $φ$ was arbitrary, (i) holds.

3.1F2F4givenstep 1.2step 2.1∎

Identification of the trace. Fix $ψ \in C_{c}^{\infty} (R^{n})$ and $β \in C_{c}^{\infty} ((- \infty, T))$ with $β (0) = 1$ . The pointwise equation from step 1.2, integrated on a box times $(δ, τ)$ containing the positive-time support of $ψ β$ , gives the calculation of steps 1.1 and 2.1 with trace ${\overset{ˉ}{u}}_{0}$ . Hence $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla φ) + \int {\overset{ˉ}{u}}_{0} ψ = 0$ for $φ = ψ β$ . Subtract the given weak identity, whose bottom term is $\int u_{0} ψ$ , to get $\int ({\overset{ˉ}{u}}_{0} - u_{0}) ψ = 0$ . By [F4], ${\overset{ˉ}{u}}_{0} = u_{0}$ almost everywhere.

PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Characteristics and the Riccati equation for the spatial derivative

Statement

Let $T > 0$ , $f \in C^{2} (R)$ , and let $u \in C^{1} (R \times (0, T))$ be a classical solution of $u_{t} + f (u)_{x} = 0$ (Scalar conservation laws, fluxes and Cauchy data, $C^{k}$ maps and multi-index derivative notation in Euclidean space).

(i) Along every characteristic $x (t)$ solving $\dot{x} = f^{'} (u (x (t), t))$ , the value $u (x (t), t)$ is constant.

(ii) If in addition $u \in C^{2} (R \times (0, T))$ , then $p (t) = u_{x} (x (t), t)$ satisfies along each characteristic $\dot{p} (t) = - f^{''} (u (x (t), t)) p (t)^{2} .$ Consequently, if $f^{''} \geq 0$ on the range of $u$ , then $p$ is nonincreasing along characteristics. More precisely, fix $t_{0} \in (0, T)$ and a characteristic through $(x_{0}, t_{0})$ , and set $p_{0} = u_{x} (x_{0}, t_{0})$ . If $p_{0} < 0$ and $f^{''} (u (x_{0}, t_{0})) \geq κ > 0$ , then, as long as the classical solution exists along that characteristic, $p (t) = \frac{p_{0}}{1 + f^{''} (u (x_{0}, t_{0})) p_{0} (t - t_{0})} .$ If the solution exists along this characteristic up to that time, its derivative tends to $- \infty$ at $t_{*} = t_{0} + 1 / (f^{''} (u (x_{0}, t_{0})) ∣ p_{0} ∣) \leq t_{0} + 1 / (κ ∣ p_{0} ∣)$ ; hence a $C^{2}$ solution cannot persist through $t_{*}$ along this characteristic (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Linear transport equations and their characteristic flow).

Facts & Assumptions

Given: $T > 0$ , $f \in C^{2} (R)$ , a classical solution $u \in C^{1} (R \times (0, T))$ of $u_{t} + f (u)_{x} = 0$ , together with a characteristic $t \mapsto x (t)$ solving $\dot{x} = f^{'} (u (x (t), t))$ ; in part (ii) additionally $u \in C^{2} (R \times (0, T))$ and $p = u_{x}$ .

[F1]

A classical solution satisfies $u_{t} + (f (u))_{x} = 0$ pointwise; since $f \in C^{2} \subset C^{1}$ and $u \in C^{1}$ , the composition $f (u)$ is $C^{1}$ with $(f (u))_{x} = f^{'} (u) u_{x}$ (Scalar conservation laws, fluxes and Cauchy data, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F2]

For the transport field $a (x, t) : = f^{'} (u (x, t))$ , which is $C^{1}$ because $f^{'}$ is $C^{1}$ and $u$ is $C^{1}$ , the solution $u$ satisfies $u_{t} + a \cdot u_{x} = 0$ , and the curve $x (\cdot)$ is a characteristic of that transport equation, so $\frac{d}{d t} u (x (t), t) = u_{t} (x (t), t) + u_{x} (x (t), t) \dot{x} (t)$ (Linear transport equations and their characteristic flow, A transport equation restricts to a linear ODE along each characteristic).

[F3]

Sums, scalar multiples and products of differentiable functions are differentiable, with the usual sum and product rules; for $u \in C^{2}$ and $p = u_{x}$ , the functions $p$ and $f^{'} (u) p$ are $C^{1}$ , while $p_{t}$ , $p_{x}$ , and $f^{''} (u)$ are continuous, so $p_{t} + f^{'} (u) p_{x} + f^{''} (u) p^{2}$ is continuous (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder). Equality $u_{t x} = u_{x t}$ is supplied by Clairaut--Schwarz theorem for continuous second partial derivatives.

[F4]

A continuous field locally Lipschitz in its state has a unique local and maximal ODE solution (Picard-Lindelöf local existence and uniqueness for first-order systems, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval). The field $f^{'} (u (x, t))$ is locally state-Lipschitz because its $x$ derivative $f^{''} (u) u_{x}$ is continuous and bounded on compact boxes.

Proof

technique · direct

1.1F1F2

Part (i). With $a (x, t) = f^{'} (u (x, t))$ , [F2] gives $\frac{d}{d t} u (x (t), t) = u_{t} + u_{x} \dot{x}$ along the characteristic. Substituting the characteristic ODE $\dot{x} = f^{'} (u (x (t), t))$ and then the pointwise equation of [F1] gives $\frac{d}{d t} u (x (t), t) = u_{t} + f^{'} (u) u_{x} = u_{t} + (f (u))_{x} = 0$ . Hence $t \mapsto u (x (t), t)$ is constant along every characteristic.

1.2F1F3

Part (ii): differentiation in $x$ . Assume now $u \in C^{2}$ , so that $p = u_{x}$ is $C^{1}$ and $f^{'} (u) p$ is $C^{1}$ by [F1] and [F3]. Differentiating the pointwise equation $u_{t} + (f (u))_{x} = 0$ in $x$ gives $p_{t} + (f^{'} (u) p)_{x} = 0$ , and the product and chain rules give $(f^{'} (u) p)_{x} = f^{''} (u) u_{x} p + f^{'} (u) p_{x} = f^{''} (u) p^{2} + f^{'} (u) p_{x}$ , so that $p_{t} + f^{'} (u) p_{x} + f^{''} (u) p^{2} = 0$ .

2.1F2step 1.2

The Riccati equation along characteristics. Along the characteristic of step 1.1, [F2] applied to the $C^{1}$ function $p$ gives $\dot{p} = p_{t} + p_{x} \dot{x} = p_{t} + f^{'} (u) p_{x}$ . By step 1.2 this equals $- f^{''} (u (x (t), t)) p^{2}$ , which is the asserted Riccati equation.

3.1step 1.1step 2.1

Monotonicity. By step 1.1, $u (x (t), t) \equiv u_{*}$ is constant along the characteristic, so along that curve $f^{''} (u (x (t), t)) = f^{''} (u_{*})$ is a constant $c$ and step 2.1 reads $\dot{p} = - c p^{2}$ . If $f^{''} \geq 0$ on the range of $u$ , then $c \geq 0$ , so $\dot{p} \leq 0$ and $p$ is nonincreasing along the characteristic.

4.1step 3.1F4algebra

Exact Riccati solution. Suppose $p_{0} = p (t_{0}) < 0$ and $c : = f^{''} (u (x_{0}, t_{0})) \geq κ > 0$ ; by step 3.1 the constant is $c = f^{''} (u_{*})$ along the whole characteristic. The scalar ODE $\dot{p} = - c p^{2}$ has a locally Lipschitz right-hand side, so uniqueness in [F4] and the zero solution imply that $p$ cannot reach zero on its interval of existence. Since $p (t_{0}) < 0$ , it remains negative there. Hence one has $\frac{d}{d t} (\frac{1}{p}) = - \dot{p} / p^{2} = c$ by step 3.1, hence $\frac{1}{p (t)} = \frac{1}{p_{0}} + c (t - t_{0})$ and therefore $p (t) = \frac{p_{0}}{1 + c p_{0} (t - t_{0})} = \frac{p_{0}}{1 - c ∣ p_{0} ∣ (t - t_{0})}$ .

5.1step 1.1step 4.1F3F4∎

Blow-up and the persistence bound. Since $u (x (t), t) \equiv u_{*}$ by step 1.1, the speed $\dot{x} = f^{'} (u_{*})$ is constant, so the characteristic is the straight line $x (t) = x_{0} + f^{'} (u_{*}) (t - t_{0})$ , on its maximal interval. If that interval ended at an interior time $t_{1} \in (0, T)$ , the straight line would have a finite endpoint $x_{1}$ ; continuity would give $u (x_{1}, t_{1}) = u_{*}$ , and [F4] would extend the characteristic with initial condition $x (t_{1}) = x_{1}$ , contradicting maximality. Thus its interval is $(0, T)$ . The denominator in step 4.1 is positive exactly for $t < t_{*} = t_{0} + 1 / (c ∣ p_{0} ∣) \leq t_{0} + 1 / (κ ∣ p_{0} ∣)$ and tends to $0$ as $t ↑ t_{*}$ , while $p_{0} < 0$ , so $p (t) \to - \infty$ . Were a $C^{2}$ solution defined on $R \times (0, T)$ with $T > t_{*}$ , then $p = u_{x}$ would be continuous, hence finite, on the compact rectangle $[- A, A] \times [t_{0}, \frac{1}{2} (t_{*} + T)]$ , with $A > 1 + ∣ x_{0} ∣ + ∣ f^{'} (u_{*}) ∣ (T - t_{0})$ , and along the straight characteristic it would equal the explicit solution of step 4.1, which is unbounded on that interval near $t_{*}$ : a contradiction. Hence the $C^{2}$ solution cannot persist through $t_{*}$ along this characteristic.

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)Open item page →

Piecewise smooth shocks and one-sided traces

Definition

Let $n \geq 1$ , $T > 0$ , and $f \in C^{1} (R; R^{n})$ (Scalar conservation laws, fluxes and Cauchy data). Let $Γ$ be a $C^{1}$ hypersurface in $Π_{T} = R^{n} \times (0, T)$ with a two-sided open neighbourhood $U \subset Π_{T}$ , so that $U ∖ Γ = U^{-} \dot{\cup} U^{+}$ for disjoint open sides $U^{\pm}$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder). Fix the unit normal $ν = (ν_{t}, ν_{x})$ on $Γ$ oriented from $U^{-}$ toward $U^{+}$ , with $ν_{t}^{2} + ∣ ν_{x} ∣^{2} = 1$ .

A piecewise $C^{1}$ weak solution near $Γ$ is a distributional weak solution $u$ of $u_{t} + {div}_{x} f (u) = 0$ on $Π_{T}$ (Distributional weak solutions of the Cauchy problem) such that at each $ζ \in Γ$ there is a neighbourhood $V_{ζ} \subset U$ and functions ${\tilde{u}}_{ζ}^{\pm} \in C^{1} (V_{ζ})$ representing $u$ on $V_{ζ} \cap U^{\pm}$ , respectively. The one-sided traces at $ζ$ are $u^{\pm} (ζ) : = {\tilde{u}}_{ζ}^{\pm} (ζ) .$ They are independent of the chosen local extensions, since continuous extensions agreeing almost everywhere on an open side agree throughout that side and at its interface points. Thus genuine traces are specified at every point of $Γ \subset U$ ; arbitrary representative values on $Γ$ do not affect the weak-solution class or these traces. Write $[u] (ζ) = u^{+} (ζ) - u^{-} (ζ), [f] (ζ) = f (u^{+} (ζ)) - f (u^{-} (ζ)) .$ A point is a shock point when $[u] (ζ) \neq 0$ .

In one space dimension, this includes a $C^{1}$ graph $x = s (t)$ , with sides $x < s (t)$ and $x > s (t)$ ; the definition is not restricted to that case, and the existence of the strong one-sided traces is part of the piecewise-smooth hypothesis, not a conclusion for arbitrary $L^{\infty}$ weak solutions.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The Rankine--Hugoniot jump condition in space--time normal form

Statement

Let $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ (Scalar conservation laws, fluxes and Cauchy data), and let $u$ be a piecewise $C^{1}$ distributional weak solution (Distributional weak solutions of the Cauchy problem) with two-sided $C^{1}$ interface $Γ$ and traces as in Piecewise smooth shocks and one-sided traces. Orient the unit space--time normal $ν = (ν_{t}, ν_{x})$ from the minus side to the plus side. Then at every $ζ \in Γ \subset U$ , $[u] (ζ) ν_{t} (ζ) + [f] (ζ) \cdot ν_{x} (ζ) = 0,$ where $[u] = u^{+} - u^{-}$ and $[f] = f (u^{+}) - f (u^{-})$ .

In one space dimension, for a graph $x = s (t)$ with minus side $x < s (t)$ and plus side $x > s (t)$ , $ν = (- s^{'} (t), 1) / \sqrt{1 + s^{'} (t)^{2}}$ , so the condition is $s^{'} (t) [u] (t) = [f] (t)$ at every graph point. If $[u] \neq 0$ in one dimension, then $s^{'} = [f] / [u]$ ; if $[u] = 0$ , then $[f] = 0$ and the relation is $0 = 0$ , with no speed constraint.

Facts & Assumptions

Given: $n \geq 1$ , a piecewise $C^{1}$ weak solution $u$ with interface $Γ \subset U$ and traces $u^{\pm}$ , a point $ζ_{0} \in Γ$ , and one-sided $C^{1}$ local extensions ${\tilde{u}}^{\pm}$ on a neighbourhood $V \subset U$ of $ζ_{0}$ .

[F1]

The weak identity reads $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla_{x} φ) = 0$ for every $φ \in C_{c}^{\infty} (Π_{T})$ , i.e. ${div}_{t, x} F (u) = 0$ in distributions, where $F (u) = (u, f (u))$ (Distributional weak solutions of the Cauchy problem, Scalar conservation laws, fluxes and Cauchy data).

[F2]

Locally about a point of a $C^{1}$ hypersurface, after permuting coordinates, a patch is a graph $z_{k} = γ (z^{'})$ over the remaining $n$ coordinates $z^{'}$ , with $γ \in C^{1}$ ; the unnormalised normal $N = e_{k} - \sum_{j \neq k} (\partial_{j} γ) e_{j}$ points from the region below the graph to the region above it, and the unit normal of [F1]'s orientation is $ν = σ N / ∣ N ∣$ with $σ = 1$ if the minus side lies below the graph and $σ = - 1$ otherwise; also $[F] = ([u], [f])$ , and a continuous function vanishing against all nonnegative smooth bumps on an open set vanishes there (Piecewise smooth shocks and one-sided traces, Explicit compactly supported smooth cutoffs).

[F3]

Iterated integration: Fubini's theorem for the product representation, the one-dimensional fundamental theorem of calculus to integrate the $z_{k}$ -derivative across the graph, and the chain rule to differentiate a moving-endpoint integral in the tangential variables (Fubini's theorem for L^1 functions on a sigma-finite product, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

Proof

technique · direct

1.1F1F2

Side extensions solve the equation classically. On each side of $Γ$ the function $u$ agrees with a $C^{1}$ extension ${\tilde{u}}^{\pm}$ ; testing away from $t = 0$ and against bumps supported in a single side, the weak identity [F1] shows that ${div}_{t, x} F ({\tilde{u}}^{\pm})$ vanishes as a distribution on that side. Since $F ({\tilde{u}}^{\pm})$ is $C^{1}$ , its divergence is continuous, and by [F2] it vanishes pointwise; consequently, for smooth compactly supported $φ$ supported in the side, ${div}_{t, x} (F (u^{\pm}) φ) = F (u^{\pm}) \cdot D φ$ .

2.1F3step 1.1

Graph computation on one side. After a permutation of coordinates, write the graph locally as $z_{k} = γ (z^{'})$ and take $φ$ supported in a box $B^{'} \times (a, b)$ in which the graph stays in $(a, b)$ . With $G_{j} = F_{j} (u^{L}) φ$ on the lower side, [F3] gives $\int_{lower} F (u^{L}) \cdot D φ = \int_{lower} div (F (u^{L}) φ) = \int_{B^{'}} G_{k}^{L} (z^{'}, γ (z^{'})) d z^{'} - \sum_{j \neq k} \int_{B^{'}} G_{j}^{L} (z^{'}, γ (z^{'})) \partial_{j} γ (z^{'}) d z^{'} = \int_{B^{'}} G^{L} (z^{'}, γ (z^{'})) \cdot N (z^{'}) d z^{'}$ , because the $k$ -derivative integrates to the trace at the graph and each tangential derivative of the moving-endpoint integral $A_{j} (z^{'}) = \int_{a}^{γ (z^{'})} G_{j} (z^{'}, r) d r$ contributes $- \partial_{j} γ G_{j}$ at the graph, the integral of $\partial_{j} A_{j}$ vanishing by compact support.

3.1F2step 1.1step 2.1

Upper side and the interface term. The same computation on the upper side, with the graph as its lower boundary, gives $\int_{upper} F (u^{R}) \cdot D φ = - \int_{B^{'}} G^{R} (z^{'}, γ (z^{'})) \cdot N (z^{'}) d z^{'}$ ; adding with step 2.1, and noting that $F (u^{+}) - F (u^{-}) = ([u], [f])$ with $u^{\pm}$ the traces from the plus and minus sides, the weak identity becomes $0 = \int_{Π_{T}} F (u) \cdot D φ = - σ \int_{B^{'}} φ (z^{'}, γ (z^{'})) [F] (z^{'}, γ (z^{'})) \cdot N (z^{'}) d z^{'}$ for every $φ$ supported in the box.

4.1F2step 3.1

Continuity and vanishing of the bracket. The function $z^{'} \mapsto [F] (z^{'}, γ (z^{'})) \cdot N (z^{'})$ is continuous, being a composition of continuous data; if it were nonzero at the point corresponding to $ζ_{0}$ , it would keep one sign on a smaller patch, and a nonnegative smooth bump supported there and positive at $ζ_{0}$ would make the integral of step 3.1 nonzero, a contradiction. Hence $[F] \cdot N = 0$ at $ζ_{0}$ .

5.1step 4.1F2∎

Normal form and the one-dimensional case. Since $ν = σ N / ∣ N ∣$ with $σ = \pm 1$ by [F2], $0 = [F] \cdot N = [F] \cdot ν σ ∣ N ∣$ , and $∣ N ∣ = ∣ e_{k} - \sum_{j \neq k} \partial_{j} γ e_{j} ∣ > 0$ , so $[F] \cdot ν = [u] ν_{t} + [f] \cdot ν_{x} = 0$ at every point of $Γ$ . For a one-dimensional graph $x = s (t)$ with minus side $x < s (t)$ , the graph function is $γ (t) = s (t)$ , so $N = (- s^{'} (t), 1)$ and $ν = (- s^{'}, 1) / \sqrt{1 + s^{' 2}}$ ; the condition becomes $(- s^{'} [u] + [f]) / \sqrt{1 + s^{' 2}} = 0$ , that is, $s^{'} [u] = [f]$ . If $[u] \neq 0$ this determines $s^{'} = [f] / [u]$ , and if $[u] = 0$ the relation reads $0 = [f]$ , so $[f] = 0$ and no speed is constrained.

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Distributional weak solutions of the Cauchy problem are not unique

Statement

Take $n = 1$ , $f (u) = u^{2}$ and $u_{0} \equiv 0$ . For every $δ > 0$ define, for $t > 0$ , $u_{δ} (t, x) = {\begin{cases} 0, & x < - δ t, \\ - δ, & - δ t < x < 0, \\ + δ, & 0 < x < δ t, \\ 0, & x > δ t . \end{cases}$ This is a bounded distributional weak solution of $u_{t} + (u^{2})_{x} = 0$ on $Π_{T}$ with strong local $L^{1}$ initial trace $u_{0} = 0$ , and it is not identically zero. At its three jumps the speeds $s = [u^{2}] / [u]$ are, respectively, $- δ$ , $0$ and $+ δ$ , so each jump coefficient $[u^{2}] - s [u]$ in the weak equation vanishes. The middle stationary jump violates the Kruzhkov entropy inequality for $k = 0$ : with $η_{0} (u) = ∣ u ∣$ and $q_{0} (u) = sgn (u) u^{2}$ its entropy production is $[q_{0}] - 0 \cdot [η_{0}] = δ^{2} - (- δ^{2}) = 2 δ^{2} > 0$ , whereas the entropy inequality requires this coefficient to be nonpositive (Kruzhkov entropy solutions). Hence $u_{δ}$ is a weak solution but not an entropy solution, and the zero solution is a distinct weak solution with the same initial data: distributional weak solutions are not unique (Distributional weak solutions of the Cauchy problem, Piecewise smooth shocks and one-sided traces).

Facts & Assumptions

Given: $n = 1$ , $f (u) = u^{2}$ , $u_{0} \equiv 0$ , $T > 0$ , $δ > 0$ , and the piecewise constant function $u_{δ}$ above, whose jump rays are $Γ_{-} = {x = - δ t}$ , $Γ_{0} = {x = 0}$ and $Γ_{+} = {x = δ t}$ in $Π_{T}$ .

[F1]

On each of the four regions the function is constant and $f$ is smooth, so $u_{δ}$ solves the equation classically there; across a jump ray $x = s (t)$ of a piecewise $C^{1}$ weak solution of $u_{t} + (u^{2})_{x} = 0$ , the distributional identity holds iff the jump coefficient $[u^{2}] - s [u]$ vanishes, where $[\cdot]$ denotes the right minus left trace across the ray (Piecewise smooth shocks and one-sided traces, Distributional weak solutions of the Cauchy problem, Scalar conservation laws, fluxes and Cauchy data).

[F2]

The Kruzhkov entropy pair for $k = 0$ is $η_{0} (u) = ∣ u ∣$ , $q_{0} (u) = sgn (u) u^{2}$ with $sgn (0) = 0$ ; an entropy solution must satisfy $\partial_{t} η_{0} (u) + \partial_{x} q_{0} (u) \leq 0$ in $D^{'} (Π_{T})$ , so across a jump ray the entropy production coefficient $[q_{0}] - s [η_{0}]$ must be nonpositive (Kruzhkov entropy solutions).

[F3]

Basic computation with the explicit states and speeds: for the ray $x = - δ t$ the left state is $0$ , the right state is $- δ$ , and the rightward speed is $s = - δ$ ; for $x = 0$ the states are $- δ$ (left) and $+ δ$ (right) with $s = 0$ ; for $x = δ t$ the states are $+ δ$ (left) and $0$ (right) with $s = δ$ ; directly $[u^{2}] - s [u] = 0$ in all three cases.

Proof

technique · direct

1.1F1F3

Weak equation. The profile is constant on its four regions. At $x = - δ t$ the right-minus-left jumps are $[u] = - δ$ , $[u^{2}] = δ^{2}$ , and $s = - δ$ , so $[u^{2}] - s [u] = δ^{2} - (- δ) (- δ) = 0$ . At $x = 0$ , $[u] = 2 δ$ , $[u^{2}] = 0$ , and $s = 0$ . At $x = δ t$ , $[u] = - δ$ , $[u^{2}] = - δ^{2}$ , and $s = δ$ , again giving zero. To verify the distributional equation, integrate $u φ_{t} + u^{2} φ_{x}$ in each region using the moving-endpoint FTC formula: each interface contributes $(s [u] - [u^{2}]) \int φ (t, s t) d t$ , which vanishes.

1.2F1given

Initial trace. For every compact $K \subseteq R$ and $0 < t < δ_{0}$ , the set where $u_{δ} (t, \cdot)$ differs from $0$ is contained in $[- δ t, δ t]$ , so $\int_{K} ∣ u_{δ} (t, x) ∣ d x \leq 2 δ^{2} t \to 0$ as $t ↓ 0$ ; hence $u_{δ}$ has the strong local $L^{1}$ initial trace $0$ . The function is bounded, hence a distributional weak solution of the Cauchy problem with datum $u_{0} \equiv 0$ in the sense of [F1].

2.1F2step 1.2

Failure of the entropy condition. At the middle ray $x = 0$ the left and right states are $- δ$ and $+ δ$ . The entropy production coefficient is $[q_{0}] - s [η_{0}]$ with $[η_{0}] = ∣ + δ ∣ - ∣ - δ ∣ = 0$ , $s = 0$ , and $[q_{0}] = q_{0} (+ δ) - q_{0} (- δ) = δ^{2} - (- δ^{2}) = 2 δ^{2} > 0$ . By [F2] the required entropy inequality fails: the distribution $\partial_{t} η_{0} (u_{δ}) + \partial_{x} q_{0} (u_{δ})$ carries the positive coefficient $2 δ^{2}$ on the ray $x = 0$ .

3.1step 1.1step 1.2step 2.1∎

Non-uniqueness. By steps 1.1 and 1.2, both $u_{δ}$ and the zero function are bounded distributional weak solutions of the same Cauchy problem with initial datum $u_{0} \equiv 0$ ; they differ on a set of positive measure for every $δ > 0$ . By step 2.1, $u_{δ}$ is not a Kruzhkov entropy solution, so the non-uniqueness occurs strictly within the class of distributional weak solutions.

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Convex entropy--entropy flux pairs

Definition

Let $n \geq 1$ , let $f ⁣ : R \to R^{n}$ be $C^{1}$ , and let $η ⁣ : R \to R$ be a finite convex locally Lipschitz function (Convex and strictly convex functions on Euclidean convex sets). An entropy flux is specified by the coordinatewise formula $q_{i} (s) = C_{i} + \int_{0}^{s} η^{'} (r) f_{i}^{'} (r) d r .$ One may use the left derivative of $η$ in this formula. Convex secant inequalities show that it is bounded and nondecreasing on compact intervals, and that it equals the ordinary derivative except at at most countably many points: assign a distinct rational to each nonempty gap between left and right slopes. A bounded monotone function is Riemann integrable, since the difference of upper and lower sums on an equal mesh is at most the mesh size times its total increase. The same holds after multiplication by the continuous $f_{i}^{'}$ : uniform continuity controls the additional oscillation in the product sums. Thus the displayed integrals exist and give locally Lipschitz $q$ . The convex secant bounds also squeeze the telescoping sum $η (b) - η (a)$ between the left and right derivative sums, proving $η (b) - η (a) = \int_{a}^{b} η^{'}$ without a choice principle. At every continuity point of $η^{'}$ , averaging the integrand over a shrinking interval gives $q^{'} (s) = η^{'} (s) f^{'} (s)$ ; the exceptional points are countable. This includes nonsmooth entropies such as $∣ s - k ∣$ . The constants $C_{i}$ are the only normalization freedom in this construction. For $η \in C^{1}$ , ordinary FTC makes $q \in C^{1}$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); second-derivative identities require $η \in C^{2}$ with $η^{''} \geq 0$ . Conversely, under Countable Choice and Dependent Choice, any locally Lipschitz $q$ satisfying $q^{'} = η^{'} f^{'}$ almost everywhere has this formula, by the fundamental theorem for absolutely continuous functions (Fundamental theorem of calculus for absolutely continuous functions, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). This converse analytic interface is separate from defining the explicit integral pairs used here.

Such a pair $(η, q)$ is an entropy--entropy flux pair. For bounded measurable $u$ its entropy inequality is $\partial_{t} η (u) + {div}_{x} q (u) \leq 0 in D^{'} (Π_{T}),$ equivalently $\int_{Π_{T}} (η (u) φ_{t} + q (u) \cdot \nabla φ) \geq 0$ for every nonnegative $φ \in C_{c}^{\infty} (Π_{T})$ (Distribution, Distributional derivative). All compositions are locally integrable on this bounded range. Adding a constant vector to $q$ leaves the inequality unchanged. The weak conservation law implies equality for the affine pairs $(η, q) = (s, f (s))$ and $(- s, - f (s))$ ; conversely their two entropy inequalities together imply that equality. The inequality for $(s, f (s))$ alone does not imply the weak equation (Scalar conservation laws, fluxes and Cauchy data).

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The viscous entropy dissipation identity

Statement

Let $n \geq 1$ , $ε > 0$ , $f \in C^{1} (R; R^{n})$ and let $u^{ε} \in C^{1, 2} (Π_{T})$ be a classical solution of the viscous conservation law $u_{t}^{ε} + {div}_{x} f (u^{ε}) = ε Δ u^{ε} in Π_{T}$ (Scalar conservation laws, fluxes and Cauchy data, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field). For every convex $η \in C^{2} (R)$ and entropy flux $q$ with $q^{'} (s) = η^{'} (s) f^{'} (s)$ , the pointwise viscous entropy balance is $η (u^{ε})_{t} + {div}_{x} q (u^{ε}) = ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2} \leq ε Δ_{x} η (u^{ε}) .$ The nonpositive term is the entropy dissipation; for fixed $ε > 0$ this is a balance with diffusion, not the first-order entropy inequality (Convex entropy--entropy flux pairs, Divergence and curl of a $C^{1}$ vector field).

Facts & Assumptions

Given: $n \geq 1$ , $ε > 0$ , $f \in C^{1} (R; R^{n})$ , a classical solution $u^{ε} \in C^{1, 2} (Π_{T})$ of the viscous conservation law, and a convex $η \in C^{2} (R)$ with entropy flux $q$ , $q^{'} = η^{'} f^{'}$ .

[F1]

Chain rules for a $C^{1}$ function of a $C^{1, 2}$ function: the composition $η (u^{ε})$ is $C^{1, 2}$ , while $q (u^{ε})$ and $f (u^{ε})$ are $C^{1}$ , with $\partial_{t} η (u^{ε}) = η^{'} (u^{ε}) u_{t}^{ε}$ , and ${div}_{x} q (u^{ε}) = q^{'} (u^{ε}) \cdot \nabla_{x} u^{ε} = η^{'} (u^{ε}) f^{'} (u^{ε}) \cdot \nabla_{x} u^{ε}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Convex entropy--entropy flux pairs).

[F2]

Laplacian of a composition: $Δ_{x} η (u^{ε}) = η^{'} (u^{ε}) Δ u^{ε} + η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ , obtained by applying the chain rule and the product rule to the components $\partial_{i} (η (u^{ε})) = η^{'} (u^{ε}) \partial_{i} u^{ε}$ and summing in $i$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

technique · direct

1.1F1F2

By [F1] the two left-hand terms are $η (u^{ε})_{t} = η^{'} (u^{ε}) u_{t}^{ε}$ and ${div}_{x} q (u^{ε}) = η^{'} (u^{ε}) f^{'} (u^{ε}) \cdot \nabla_{x} u^{ε}$ , and by [F2] the diffusion term is $Δ_{x} η (u^{ε}) = η^{'} (u^{ε}) Δ u^{ε} + η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ .

2.1step 1.1F1algebra

Multiplying the viscous equation pointwise by $η^{'} (u^{ε})$ and adding the second identity of step 1.1 gives $η (u^{ε})_{t} + {div}_{x} q (u^{ε}) = η^{'} (u^{ε}) (u_{t}^{ε} + {div}_{x} f (u^{ε}) - ε Δ u^{ε}) + ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2} = ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ , which is the asserted balance.

3.1step 2.1F2∎

The convexity assumption gives $η^{''} \geq 0$ , so the dissipation term $- ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ is nonpositive pointwise and the balance implies the stated inequality; the term $ε Δ_{x} η (u^{ε})$ may change sign and cannot be dropped pointwise for fixed $ε > 0$ .

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Kruzhkov entropy solutions

Definition

Let $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ , and $Π_{T} = R^{n} \times (0, T)$ . The unknown and initial datum are equivalence classes $[u] \in L^{\infty} (Π_{T})$ and $[u_{0}] \in L^{\infty} (R^{n}) \cap L_{l o c}^{1} (R^{n})$ , where equality is Lebesgue-a.e. (The space $L^{p} (μ)$ as the quotient by null functions, A locally integrable function on $R^{n}$ ).

For $k \in R$ define the Kruzhkov entropy pair $η_{k} (s) = ∣ s - k ∣, q_{k} (s) = sgn (s - k) (f (s) - f (k)),$ with $sgn (0) = 0$ (Absolute value in an ordered field). Then $η_{k}$ is convex and $q_{k}^{'} = η_{k}^{'} f^{'}$ almost everywhere, so $(η_{k}, q_{k})$ is a locally Lipschitz convex entropy--entropy flux pair (Convex entropy--entropy flux pairs).

The class $[u]$ is a Kruzhkov entropy solution of $u_{t} + {div}_{x} f (u) = 0$ with initial trace $[u_{0}]$ if:

(i) for every $k \in R$ and every nonnegative $φ \in C_{c}^{\infty} (Π_{T})$ , $\int_{Π_{T}} (η_{k} (u) φ_{t} + q_{k} (u) \cdot \nabla_{x} φ) d x d t \geq 0,$ that is, $\partial_{t} η_{k} (u) + {div}_{x} q_{k} (u) \leq 0$ in $D^{'} (Π_{T})$ ; and

(ii) for every compact $K \subseteq R^{n}$ , $\lim_{δ ↓ 0} {ess sup}_{0 < t < \min {δ, T}} \int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x = 0.$

The integral in (i) is independent of the representative because its integrand is unchanged almost everywhere. For (ii), Fubini's theorem gives locally integrable spatial sections for almost every $t$ (Fubini's theorem for L^1 functions on a sigma-finite product); the displayed slice integral is defined for those times and its essential supremum ignores the exceptional null set. If $u$ or $u_{0}$ is changed on a null set in its respective space, Fubini's theorem shows that the slice-integral function changes only for a null set of times, so the trace condition is well defined on the equivalence classes: this is the strong local $L^{1}$ initial trace.

Taking $k$ above and below the essential range of $u$ makes $η_{k}$ equal $k - u$ and $u - k$ , whose $t$ - and $x$ -derivatives cancel the constant terms against compactly supported test functions, so the entropy inequalities imply the weak conservation law of Distributional weak solutions of the Cauchy problem tested against nonnegative test functions, hence by linearity against all test functions.

To recover the Cauchy boundary term, apply the interior weak identity to $φ (t, x) ζ (t / δ)$ , where $ζ$ is smooth and nondecreasing, $ζ = 0$ on $(- \infty, 1 / 2]$ and $ζ = 1$ on $[1, \infty)$ . For $δ > 0$ this product is supported away from $t = 0$ , so it is an admissible interior test. The term containing $ζ^{'} / δ$ converges to $\int u_{0} (x) φ (0, x) d x$ by (ii), while the other terms converge on the compact support. This proves the full weak formulation with initial datum $u_{0}$ , rather than only its interior equation.

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The viscous scalar Cauchy problem with smooth data has a global classical solution

Statement

Assume Countable Choice (CC) for the heat-kernel and $L^{1}$ interfaces below. Let $n \geq 1$ , $ε > 0$ , let $f ⁣ : R \to R^{n}$ be $C^{2}$ with $f (0) = 0$ , and let $u_{0} \in C_{c}^{\infty} (R^{n})$ . For every $T > 0$ there is a mild solution $u$ of $u_{t} + {div}_{x} f (u) = ε Δ u on R^{n} \times (0, T), u (x, 0) = u_{0} (x),$ with $u \in C ([0, T]; C_{b} (R^{n})) \cap C ([0, T]; L^{1} (R^{n})) \cap C^{1, 2} (R^{n} \times (0, T))$ . The solution is global in the sense that these solutions are compatible on finite time intervals; for every $t \in [0, T]$ , $\sup_{x \in R^{n}} u (t, x) \leq \sup_{x} u_{0} (x), \inf_{x \in R^{n}} u (t, x) \geq \inf_{x} u_{0} (x) .$ In particular it is bounded and remains in the initial range. No uniqueness beyond the constructed mild solution is asserted.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $ε > 0$ , $f \in C^{2} (R; R^{n})$ with $f (0) = 0$ , $u_{0} \in C_{c}^{\infty} (R^{n})$ , and $T > 0$ .

[F1]

The heat evolution $H_{t}$ is the convolution with the heat kernel $Γ$ : $H_{t} g = Γ (\cdot, t) * g$ is defined for $g \in L^{p}$ with $∥ H_{t} g ∥_{p} \leq ∥ g ∥_{p}$ , and $Γ (\cdot, t)$ has unit mass, is strictly positive, is $C^{\infty}$ with $\partial_{t} Γ = Δ Γ$ , and satisfies the Gaussian derivative bounds $∣ D^{α} Γ (x, t) ∣ \leq C_{n, α} t^{- (∣ α ∣ + n) / 2} e^{- ∣ x ∣^{2} / (8 t)}$ (The heat evolution $H_{t}$ of initial data, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F2]

For every multi-index $α$ and $1 \leq p \leq q \leq \infty$ there is $C_{α, p, q}$ with $∥ D^{α} H_{t} g ∥_{q} \leq C_{α, p, q} t^{- ∣ α ∣ / 2 - \frac{n}{2} (1 / p - 1 / q)} ∥ g ∥_{p}$ for all $g \in L^{p}$ and $t > 0$ ; moreover the spatial and time derivatives pass through heat convolution for positive time, and Young's inequality bounds convolutions (Spatial derivative estimates for the heat flow, $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow, Spatial and time derivatives pass through heat convolution for positive time, Young's convolution inequality under Countable Choice).

[F3]

$H_{t} u_{0}$ is bounded and uniformly continuous with $∥ H_{t} u_{0} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ , and $C ([0, τ]; C_{b} (R^{n}))$ with the supremum norm is complete: a uniformly Cauchy sequence of bounded functions converges pointwise in $R$ , its Cauchy bound then gives uniform convergence and boundedness of the limit; uniform limits preserve spatial continuity. Applying the same argument to continuous paths with values in this complete $C_{b}$ space gives a uniform limit continuous in time (the triangle inequality with one approximating path proves continuity). Thus this path space is complete, the uniform limit of continuous functions is continuous, and the Banach fixed point theorem applies to a contraction of a nonempty complete metric space (The heat Cauchy problem for bounded uniformly continuous data, The uniform limit of continuous real-valued functions on a metric space is continuous, Closed subspaces of complete metric spaces are complete; the converse under countable choice, A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point).

[F4]

Differentiable calculus: the chain rule, the sum and product rules, and the Laplacian of a $C^{2}$ function; parabolic cylinders and their parabolic boundary are as defined for the maximum principle (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Parabolic cylinder and parabolic boundary).

[F5]

Convergence tools: dominated convergence, Riesz--Fischer completeness of $L^{p}$ , and the $L^{p}$ class convention; the extreme value theorem on compact sets and Heine--Borel in $R^{n}$ (Dominated convergence, Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , The space $L^{p} (μ)$ as the quotient by null functions, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F6]

Write $B U C (R^{n})$ for the bounded uniformly continuous functions. The Gaussian kernels form an $L^{1}$ approximate identity, so $H_{ε t} g \to g$ in $L^{1}$ for $g \in L^{1}$ by the approximate-identity theorem. For $g \in B U C$ , define $ω_{g} (ρ) : = \sup_{∣ z ∣ \leq ρ} ∥ g (\cdot - z) - g ∥_{\infty}$ , which tends to zero with $ρ$ . Unit mass and the Gaussian tail give $∥ H_{ε t} g - g ∥_{\infty} \leq ω_{g} (ρ) + 2 ∥ g ∥_{\infty} \int_{∣ z ∣ > ρ} Γ (z, ε t) d z,$ so $H_{ε t} g \to g$ uniformly as $t ↓ 0$ . The Gaussian convolution identity $Γ_{t} * Γ_{s} = Γ_{t + s}$ follows by completing the square in its integrand and using Gaussian unit mass. The contraction and this identity give strong continuity of $t \mapsto H_{ε t} g$ in both $L^{1}$ and $B U C$ on $[0, τ]$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ). The product-space rearrangements use Fubini's theorem for L^1 functions on a sigma-finite product, kernel time integrations use The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , and the flux Lipschitz bound uses The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ .

Proof

technique · direct

1.1F1F2F6

Setup and kernel estimates. Put $M_{0} : = ∥ u_{0} ∥_{\infty}$ , $R : = 2 M_{0} + 1$ and $L : = \sup_{∣ s ∣ \leq R} ∣ f^{'} (s) ∣ < \infty$ . Since $f (0) = 0$ and $f$ is $C^{1}$ , the mean value theorem gives $∣ f (s) ∣ \leq L ∣ s ∣$ for $∣ s ∣ \leq R$ . By [F2] there are constants $C_{1}, C_{2}$ with $∥ \nabla H_{σ} g ∥_{p} \leq C_{1} σ^{- 1 / 2} ∥ g ∥_{p}$ for $p \in {1, \infty}$ . The Gaussian first moment gives $∥ H_{σ} g - g ∥_{\infty} \leq C_{2} σ^{1 / 2} ∥ \nabla g ∥_{\infty}$ for $g \in C^{1} \cap L^{\infty}$ with $\nabla g$ bounded. Also, $∥ \nabla Γ_{σ} (\cdot + h) - \nabla Γ_{σ} ∥_{1} \leq \min {2 C_{1} σ^{- 1 / 2}, C_{2} ∣ h ∣ σ^{- 1}}$ and $∥ Γ_{σ} (\cdot + h) - Γ_{σ} ∥_{1} \leq \min {2, C_{2} ∣ h ∣ σ^{- 1 / 2}}$ for every $h \in R^{n}$ and $σ > 0$ . For the scaled kernel, $∥ \partial_{σ} \nabla Γ_{ε σ} ∥_{1} \leq C_{ε} σ^{- 3 / 2}$ , so for $Δ > 0$ and $σ > 0$ , $∥ \nabla Γ_{ε (σ + Δ)} - \nabla Γ_{ε σ} ∥_{1} \leq C_{ε} \min {σ^{- 1 / 2}, Δ σ^{- 3 / 2}} .$

2.1F1F3F6step 1.1

A local mild solution for bounded uniformly continuous $L^{1}$ data. Fix a starting datum $g \in B U C (R^{n}) \cap L^{1} (R^{n})$ with $∥ g ∥_{\infty} \leq M_{0}$ . Choose $τ_{0} \in (0, T]$ with $q : = 2 C_{1} L \sqrt{τ_{0} / ε} \leq \frac{1}{2}$ and let $X_{τ_{0}} : = C ([0, τ_{0}]; C_{b} (R^{n}))$ with the supremum norm (Closed subspaces of complete metric spaces are complete; the converse under countable choice). For $v \in X_{τ_{0}}$ with $∥ v ∥_{X_{τ_{0}}} \leq R$ define $(T_{g} v) (t) : = H_{ε t} g - \int_{0}^{t} \nabla H_{ε (t - s)} * f (v (s)) d s .$ By [F6], $t \mapsto H_{ε t} g$ is continuous in the supremum norm at $t = 0$ ; by [F1] the Duhamel term is continuous in $X_{τ_{0}}$ , since its integrand is norm continuous for $s < t$ and has the integrable majorant $C_{1} (ε (t - s))^{- 1 / 2} ∥ f (v (s)) ∥_{\infty}$ . The bounds $∥ H_{ε t} g ∥_{\infty} \leq M_{0}$ and $∥ f (v (s)) ∥_{\infty} \leq L R$ give $∥ T_{g} v ∥_{X_{τ_{0}}} \leq M_{0} + q R \leq R, ∥ T_{g} u - T_{g} v ∥_{X_{τ_{0}}} \leq q ∥ u - v ∥_{X_{τ_{0}}} .$ Thus $T_{g}$ is a contraction of the closed radius- $R$ ball; the Banach fixed point theorem gives a unique fixed point $u \in X_{τ_{0}}$ satisfying $u (t) = H_{ε t} g - \int_{0}^{t} \nabla H_{ε (t - s)} * f (u (s)) d s$ . The original datum $u_{0} \in C_{c}^{\infty}$ satisfies these assumptions.

2.2F1F2step 1.1

Hölder regularity on strips. Fix $0 < δ < τ_{0}$ and let $t, t^{'} \in [δ, τ_{0}]$ . Since $f (u)$ is bounded with $∥ f (u) ∥_{\infty} \leq L R$ , the mild identity and the kernel bounds of step 1.1 give, for every $α \in (0, 1)$ , $∥ u (t, \cdot + h) - u (t, \cdot) ∥_{\infty} \leq C_{δ} ∣ h ∣^{α}, ∥ u (t, \cdot) - u (t^{'}, \cdot) ∥_{\infty} \leq C_{δ} ∣ t - t^{'} ∣^{1 / 2} .$ For the spatial estimate, $∥ \nabla H_{ε t} g ∥_{\infty} \leq C_{ε, δ} ∥ g ∥_{\infty}$ on the strip. The Duhamel term is bounded by $L R \int_{0}^{t} \min {C_{ε} σ^{- 1 / 2}, C_{ε} ∣ h ∣ σ^{- 1}} d σ$ , with $t \leq τ_{0}$ ; splitting at $σ = ∣ h ∣^{2}$ gives at most $C_{ε} (∣ h ∣ + ∣ h ∣ \log_{+} (τ_{0} / ∣ h ∣^{2})) \leq C_{ε, τ_{0}, α} ∣ h ∣^{α}$ for $0 < ∣ h ∣ \leq 1$ , where $\log_{+} (r) : = \max {\log r, 0}$ . Larger $∣ h ∣$ are covered by boundedness of $u$ . For the temporal estimate assume $t = t^{'} + Δ$ with $Δ > 0$ . The initial heat term is Lipschitz in $t$ on $[δ, τ_{0}]$ by the positive-time estimate for $\partial_{t} H_{ε t} g = ε Δ H_{ε t} g$ . On the common Duhamel interval, with $σ = t^{'} - s$ , the kernel difference is a time difference and satisfies $∥ \nabla Γ_{ε (σ + Δ)} - \nabla Γ_{ε σ} ∥_{1} \leq C_{ε} \min {σ^{- 1 / 2}, Δ σ^{- 3 / 2}} .$ Splitting its integral at $σ = Δ$ gives $\int_{0}^{t^{'}} ∥ \nabla Γ_{ε (σ + Δ)} - \nabla Γ_{ε σ} ∥_{1} d σ \leq C_{ε} \sqrt{Δ}$ . The remaining time slab has norm at most $L R \int_{t^{'}}^{t} C_{1} (ε (t - s))^{- 1 / 2} d s \leq 2 C_{1} L R \sqrt{Δ / ε}$ . Thus both estimates hold with $C_{δ}$ depending only on $δ, τ_{0}, ε, n, ∥ g ∥_{\infty}, L R$ , and $u$ is continuous in $(t, x)$ for $t > 0$ .

3.1F2F5F6step 2.1

$L^{1}$ continuity and identification of the limits. For the arbitrary starting datum $g \in B U C \cap L^{1}$ of step 2.1, take Picard iterates $u^{(0)} : = g$ , $u^{(k + 1)} : = T_{g} u^{(k)}$ . By [F6], $u^{(0)} \in C ([0, τ_{0}]; L^{1})$ . If $u^{(k)} \in C ([0, τ_{0}]; L^{1})$ , the integral defining $u^{(k + 1)}$ is defined by norm limits of Riemann sums on truncated intervals $s \leq t - η$ . Completeness supplies these limits, and the bound on the omitted interval is at most $2 C_{1} L ∥ u^{(k)} ∥_{C_{t} L^{1}} \sqrt{η / ε}$ , which tends to zero. Its integrand has integrable majorant $C_{1} (ε s)^{- 1 / 2} L ∥ u^{(k)} ∥_{C_{t} L^{1}}$ , so $u^{(k + 1)}$ is $L^{1}$ -continuous and $∥ u^{(k + 1)} ∥_{C_{t} L^{1}} \leq ∥ g ∥_{1} + q ∥ u^{(k)} ∥_{C_{t} L^{1}} \leq 2 ∥ g ∥_{1}$ . Successive differences obey $∥ u^{(k + 1)} - u^{(k)} ∥_{C_{t} L^{1}} \leq q ∥ u^{(k)} - u^{(k - 1)} ∥_{C_{t} L^{1}}$ , so the iterates converge in the complete space $C ([0, τ_{0}]; L^{1})$ (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ ) to some $w$ . They also converge uniformly on $[0, τ_{0}] \times R^{n}$ to the fixed point $u$ of step 2.1. On every bounded ball $B_{m}$ , this uniform convergence implies convergence to $u$ in $C ([0, τ_{0}]; L^{1} (B_{m}))$ , while the global $L^{1}$ convergence gives convergence to $w$ in the same local space. Uniqueness of the local $L^{1}$ limit yields $u = w$ a.e. on every $B_{m}$ , hence a.e. on $R^{n}$ ; thus $u \in C ([0, τ_{0}]; L^{1})$ and $∥ u (t) ∥_{1} \leq 2 ∥ g ∥_{1}$ .

3.2F1F2step 2.1

The equation in distributions. Testing the mild identity and using Fubini (the bound $(t - s)^{- 1 / 2}$ is integrable on each finite time triangle) gives $\int_{Π} (u φ_{t} + f (u) \cdot \nabla φ) + \int g φ (\cdot, 0) = - ε \int_{Π} u Δ φ .$ Indeed the initial heat term pairs as $- \int g φ (\cdot, 0)$ against $φ_{t} + ε Δ φ$ , and the divergence Duhamel term pairs as $- \int f (u) \cdot \nabla φ$ against the same expression, by integrating $\partial_{t} H_{ε (t - s)} = ε Δ H_{ε (t - s)}$ in $t$ . Thus $u_{t} + div f (u) = ε Δ u$ distributionally, with datum $g$ .

3.3F1F2F4step 1.1step 2.2

Hölder continuity and boundedness of the spatial gradient. Fix $0 < δ_{0} < δ < τ_{0}$ and use the mild identity restarted at $δ_{0}$ : $u (t) = H_{ε (t - δ_{0})} u (δ_{0}) - div W (t), W (t) : = \int_{δ_{0}}^{t} H_{ε (t - s)} f (u (s)) d s .$ By step 2.2, $f (u)$ is spatially $C^{α}$ and temporally $C^{α / 2}$ on each positive strip (the $C^{1 / 2}$ bound implies the weaker $C^{α / 2}$ bound). Componentwise, differentiation of the divergence heat potential gives $\partial_{k} u (t, x) = \partial_{k} H_{ε (t - δ_{0})} u (δ_{0}, x) - \sum_{i} \int_{0}^{t - δ_{0}} \int_{R^{n}} D_{k i} Γ_{ε σ} (z) (f_{i} (u (t - σ, x - z)) - f_{i} (u (t, x))) d z d σ,$ where the frozen value is subtracted using $\int D_{k i} Γ_{ε σ} = 0$ . Gaussian scaling gives $∥ D^{2} Γ_{ε σ} ∥_{1} \leq C_{ε} σ^{- 1}$ , $∥ \nabla D^{2} Γ_{ε σ} ∥_{1} \leq C_{ε} σ^{- 3 / 2}$ , and $∥ \partial_{σ} D^{2} Γ_{ε σ} ∥_{1} \leq C_{ε} σ^{- 2}$ . With the parabolic $C^{α, α / 2}$ modulus of $f (u)$ , a spatial increment $h$ is bounded after splitting at $σ = ∣ h ∣^{2}$ by $C \int_{0}^{∣ h ∣^{2}} σ^{- 1 + α / 2} d σ + C ∣ h ∣ \int_{∣ h ∣^{2}}^{τ_{0} - δ_{0}} σ^{- 3 / 2 + α / 2} d σ \leq C ∣ h ∣^{α},$ and a time increment $Δ = ∣ t - t^{'} ∣$ is bounded after splitting at $σ = Δ$ by $C \int_{0}^{Δ} σ^{- 1 + α / 2} d σ + C Δ \int_{Δ}^{τ_{0} - δ_{0}} σ^{- 2 + α / 2} d σ \leq C Δ^{α / 2} .$ If either split point exceeds the finite integration horizon, the same bounds follow by increasing $C$ . The heat initial term is smooth with bounded derivatives on $t \geq δ$ . Thus for each $0 < α < 1$ , $[\nabla u]_{C_{x}^{α} (R^{n} \times [δ, τ_{0}))} + [\nabla u]_{C_{t}^{α / 2} (R^{n} \times [δ, τ_{0}))} \leq C_{δ_{0}, δ} .$ These are seminorm bounds; in particular they establish continuity of $\nabla u$ before any absolute bound is used. The absolute gradient bound follows from boundedness of $u$ . If $\nabla u (t, x) \neq 0$ , put $e = \nabla u (t, x) / ∣ \nabla u (t, x) ∣$ . Along the unit segment $x + s e$ , $0 \leq s \leq 1$ , the fundamental theorem of calculus and the spatial seminorm $M_{δ} : = [\nabla u]_{C_{x}^{α}}$ give $∣ \nabla u (t, x) ∣ \leq ∣ u (t, x + e) - u (t, x) ∣ + \int_{0}^{1} ∣ \nabla u (t, x + s e) - \nabla u (t, x) ∣ d s \leq 2 R + \frac{M_{δ}}{1 + α} .$ Thus $\nabla u$ is bounded on the positive-time strip. Since $f^{''}$ is bounded on $[- R, R]$ , $f^{'} (u)$ has the same parabolic Hölder regularity as $u$ there; hence $h : = - f^{'} (u) \cdot \nabla u$ is bounded and belongs to $C^{α, α / 2} (R^{n} \times [δ, τ_{0}))$ .

4.1F1F4F5step 2.1step 3.3

Heat-potential cancellation and $C^{1, 2}$ . Fix $0 < δ < τ_{0}$ , use the bounds of step 3.3 on a slightly larger positive strip, and restart the heat equation at $δ$ . Then $u (t) = H_{ε (t - δ)} u (δ) + w (t), w (t, x) : = \int_{δ}^{t} H_{ε (t - s)} h (s) (x) d s .$ For $σ > 0$ , $\int_{R^{n}} D_{i j} Γ_{ε σ} (z) d z = 0$ , so cancellation gives $D_{i j} w (t, x) = \int_{0}^{t - δ} \int_{R^{n}} D_{i j} Γ_{ε σ} (z) (h (t - σ, x - z) - h (t, x)) d z d σ .$ The parabolic Hölder bound from step 3.3 yields $∣ h (t - σ, x - z) - h (t, x) ∣ \leq C (∣ z ∣^{α} + σ^{α / 2})$ . Gaussian scaling therefore gives $\int_{R^{n}} ∣ D^{2} Γ_{ε σ} (z) ∣ (∣ z ∣^{α} + σ^{α / 2}) d z \leq C_{ε, n, α} σ^{- 1 + α / 2},$ which is integrable at $σ = 0$ . Truncating the heat-potential integral at $σ = η > 0$ , differentiating, and letting $η ↓ 0$ with this integrable dominator shows that $D_{i j} w$ exists and is continuous; the heat-kernel identity also gives $w_{t} - ε Δ w = h$ in distributions. Since $h$ and $Δ w$ are continuous, this identity gives a continuous classical time derivative. The first term $H_{ε (t - δ)} u (δ)$ is smooth for $t > δ$ , so $u \in C^{1, 2} (R^{n} \times (δ, τ_{0}))$ and satisfies $u_{t} = ε Δ u - f^{'} (u) \cdot \nabla u$ , hence $u_{t} + {div}_{x} f (u) = ε Δ u$ pointwise. Since $δ > 0$ is arbitrary, $u \in C^{1, 2} (R^{n} \times (0, τ_{0}))$ .

5.1F3F4step 2.1step 4.1

The range bound by a barrier. Let $a < b$ in $[0, τ_{0})$ and put $M_{a} : = \sup_{x} u (a, x)$ . Write $B : = f^{'} (u (t, x))$ , which is bounded on $[a, b] \times R^{n}$ by $L$ . For $δ_{1} > 0$ and $λ > 0$ set $Φ (t, x) : = δ_{1} e^{λ (t - a)} (1 + ∣ x ∣^{2})$ ; then, using steps 4.1 and [F4], $(\partial_{t} - ε Δ + B \cdot \nabla) Φ = δ_{1} e^{λ (t - a)} (λ (1 + ∣ x ∣^{2}) - 2 ε n + 2 B \cdot x) > 0$ for $λ$ large enough, because $∣ B ∣ \leq L$ and $- ε Δ (1 + ∣ x ∣^{2}) = - 2 ε n$ . The function $z : = u - M_{a} - Φ$ is negative at $t = a$ and, by step 2.1, $∣ u ∣ \leq R$ while $Φ \geq δ_{1} (1 + ∣ x ∣^{2})$ , so $z$ is negative on the lateral boundary of every sufficiently large ball intersected with $[a, b]$ ; if $z$ had a positive maximum in the closed cylinder, then at that point $z_{t} \geq 0$ , $D z = 0$ , $Δ z \leq 0$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), contradicting $(\partial_{t} - ε Δ + B \cdot \nabla) z < 0$ there. Letting $δ_{1} ↓ 0$ gives $u \leq M_{a}$ on $[a, b]$ ; applying the same argument to $- u$ , whose flux is $\tilde{f} (s) = - f (- s)$ and whose range bound is the same, gives $u \geq m_{a} : = \inf_{x} u (a, x)$ . In particular, with $a = 0$ , $\sup_{x} u (t, x) \leq \sup_{x} u_{0}$ and $\inf_{x} u (t, x) \geq \inf_{x} u_{0}$ on $[0, τ_{0})$ , the range of $u$ is contained in $I_{0} : = [\inf u_{0}, \sup u_{0}]$ .

6.1F3F6step 2.1step 2.2step 3.1step 4.1step 5.1∎

Global continuation and conclusion. For every $t_{1} > 0$ , the positive-strip spatial modulus of step 2.2 makes $u (t_{1}, \cdot)$ bounded and uniformly continuous, step 3.1 gives $u (t_{1}) \in L^{1}$ , and step 5.1 places its values in $I_{0}$ . Thus $g : = u (t_{1})$ is an admissible $B U C \cap L^{1}$ starting datum for the local fixed-point and Picard iteration of steps 2.1 and 3.1. The strong heat continuity required for this restart is [F6]: Gaussian approximate-identity convergence gives $H_{ε t} g \to g$ in the supremum norm for BUC data and in $L^{1}$ for $L^{1}$ data. With the same global constants $R, L$ , the local solution on $[t_{1}, t_{1} + τ_{0})$ agrees with $u$ at $t_{1}$ ; local uniqueness in step 2.1 makes the pieces agree on overlaps, and $∥ u (t) ∥_{\infty} \leq \max {∣ \inf u_{0} ∣, ∣ \sup u_{0} ∣} \leq R$ throughout. Iterating finitely many times covers $[0, T]$ , and the same local uniqueness shows that two solutions obtained with different terminal times $T, T^{'}$ agree on their common interval. The range bounds of step 5.1 hold on all of $[0, T]$ ; $L^{1}$ continuity holds by step 3.1 and $C^{1, 2}$ regularity on $R^{n} \times (0, T)$ by step 4.1. This is the asserted global mild solution.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Uniform L-infinity, mass and energy bounds for the viscous approximations

Statement

Assume Countable Choice (CC). Let $n \geq 1$ , $0 < ε \leq 1$ , $f \in C^{2} (R; R^{n})$ with $f (0) = 0$ , and $u_{0} \in C_{c}^{\infty} (R^{n})$ . Let $u^{ε}$ be the viscous solution constructed in The viscous scalar Cauchy problem with smooth data has a global classical solution, so that $u^{ε} \in C ([0, T]; C_{b} (R^{n})) \cap C ([0, T]; L^{1} (R^{n})) \cap C^{1, 2} (R^{n} \times (0, T))$ solves $u_{t}^{ε} + {div}_{x} f (u^{ε}) = ε Δ u^{ε}$ pointwise. Then for every $t \in [0, T]$ :

(i) $∥ u^{ε} (t) ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ ;

(ii) signed mass is conserved, $\int_{R^{n}} u^{ε} (t, x) d x = \int_{R^{n}} u_{0} (x) d x$ , while the $L^{1}$ norm is nonincreasing, $∥ u^{ε} (t) ∥_{1} \leq ∥ u_{0} ∥_{1}$ (in general it is not constant);

(iii) the integrated energy identity $\frac{1}{2} ∥ u^{ε} (t) ∥_{2}^{2} + ε \int_{0}^{t} ⁣ ⁣ \int_{R^{n}} ∣ \nabla u^{ε} ∣^{2} d x d s = \frac{1}{2} ∥ u_{0} ∥_{2}^{2},$ so that $ε \int_{0}^{T} ⁣ ⁣ \int ∣ \nabla u^{ε} ∣^{2} \leq \frac{1}{2} ∥ u_{0} ∥_{2}^{2}$ , uniformly for $0 < ε \leq 1$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < ε \leq 1$ , $f \in C^{2}$ with $f (0) = 0$ , $u_{0} \in C_{c}^{\infty}$ , the viscous solution $u^{ε}$ , a time $t \in [0, T]$ , and the constant $L_{M} : = \sup_{∣ s ∣ \leq ∥ u_{0} ∥_{\infty}} ∣ f^{'} (s) ∣$ .

[F1]

The constructed solution obeys the range bound $\sup_{x} u^{ε} (t, x) \leq \sup_{x} u_{0}$ and $\inf_{x} u^{ε} (t, x) \geq \inf_{x} u_{0}$ , has $A_{T} : = \sup_{0 \leq s \leq T} ∥ u^{ε} (s) ∥_{1} < \infty$ by its $L^{1}$ -continuous orbit, and solves $u_{t}^{ε} + div f (u^{ε}) = ε Δ u^{ε}$ pointwise (The viscous scalar Cauchy problem with smooth data has a global classical solution).

[F2]

The viscous entropy balance holds pointwise for every convex $C^{2}$ entropy: $\partial_{t} η (u^{ε}) + div q (u^{ε}) = ε Δ η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla u^{ε} ∣^{2}$ with $q^{'} = η^{'} f^{'}$ (The viscous entropy dissipation identity, Convex entropy--entropy flux pairs).

[F3]

There are smooth radial cutoffs $0 \leq χ_{R} \leq 1$ with $χ_{R} = 1$ on $B_{R}$ , $χ_{R} = 0$ outside $B_{2 R}$ , $χ_{R} ↑ 1$ as $R \to \infty$ , $∣ D χ_{R} ∣ \leq C / R$ and $∣ Δ χ_{R} ∣ \leq C / R^{2}$ : use the explicit profile $χ_{R} (x) = σ ((4 - ∣ x ∣^{2} / R^{2}) / 3)$ from the cited construction. On $0 < t < 1$ , differentiating its defining quotient gives $σ^{'} (t) = σ (t) (1 - σ (t)) (t^{- 2} + (1 - t)^{- 2}) > 0$ , so the profile is nonincreasing in radius and $χ_{R}$ increases with $R$ . The derivative scaling gives the stated gradient and Laplacian bounds (Explicit compactly supported smooth cutoffs, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F4]

Spatial integration by parts against a compactly supported smooth $χ$ follows from the one-dimensional theorem, not from a theorem on balls: enclose $supp χ$ in the interior of a box, fix all coordinates except $x_{i}$ , and apply Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives on that coordinate interval. The integrands are continuous, their Riemann and Lebesgue integrals agree by A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral, and Fubini (Fubini's theorem for L^1 functions on a sigma-finite product) integrates the identity over the remaining coordinates. Boundary terms vanish since $χ$ vanishes near the box boundary. Summing gives $\int χ div Q = - \int Q \cdot \nabla χ$ ; applying the same argument twice gives $\int χ Δ v = \int v Δ χ$ for $Q \in C^{1}$ and $v \in C^{2}$ . Dominated and monotone convergence justify the indicated cutoff and nonnegative limits (Dominated convergence, Monotone convergence for the integral, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F1

Supremum bound. Statement (i) is exactly the range bound of the construction theorem [F1]: $∥ u^{ε} (t) ∥_{\infty} \leq \max {∣ \sup_{x} u_{0} ∣, ∣ \inf_{x} u_{0} ∣} = ∥ u_{0} ∥_{\infty}$ .

1.2F1F3F4

Mass identity with cutoffs. Multiply the pointwise equation by a cutoff $χ_{R}$ of [F3], integrate first from a positive lower time, and then pass that time to zero by $C ([0, T]; L^{1})$ continuity: since $χ_{R}$ vanishes outside a compact set, integration by parts is legitimate and gives $\int u^{ε} (t) χ_{R} - \int u_{0} χ_{R} = \int_{0}^{t} ⁣ ⁣ \int (f (u^{ε}) \cdot \nabla χ_{R} + ε u^{ε} Δ χ_{R})$ ; by [F1] the right side is bounded in absolute value by $(C L_{M} R^{- 1} + ε C R^{- 2}) \int_{0}^{t} ∥ u^{ε} (s) ∥_{1} d s \leq C_{T} (R^{- 1} + ε R^{- 2})$ .

1.3F1F2F3F4

Positive-time energy identity. Put $M = ∥ u_{0} ∥_{\infty}$ . Fix $0 < s < t \leq T$ , take the convex entropy $η (r) = r^{2} / 2$ and its flux $q (r) = \int_{0}^{r} a f^{'} (a) d a$ , and integrate the balance [F2] against $χ_{R}$ over $[s, t] \times R^{n}$ . This is legitimate on each compact support because $u^{ε} \in C^{1, 2}$ for positive times. The resulting identity is $\frac{1}{2} \int ∣ u^{ε} (t) ∣^{2} χ_{R} + ε \int_{s}^{t} ⁣ ⁣ \int ∣ \nabla u^{ε} ∣^{2} χ_{R} = \frac{1}{2} \int ∣ u^{ε} (s) ∣^{2} χ_{R} + \int_{s}^{t} ⁣ ⁣ \int q (u^{ε}) \cdot \nabla χ_{R} + ε \int_{s}^{t} ⁣ ⁣ \int η (u^{ε}) Δ χ_{R} .$ On the solution range, $∣ q (u^{ε}) ∣ \leq L_{M} M ∣ u^{ε} ∣ / 2$ and $η (u^{ε}) \leq M ∣ u^{ε} ∣ / 2$ , so both cutoff errors tend to zero by [F1, F3]; the endpoint energies converge by dominated convergence. Since $χ_{R} ↑ 1$ , monotone convergence applies to the nonnegative dissipation and gives $\frac{1}{2} ∥ u^{ε} (t) ∥_{2}^{2} + ε \int_{s}^{t} ⁣ ⁣ \int ∣ \nabla u^{ε} ∣^{2} = \frac{1}{2} ∥ u^{ε} (s) ∥_{2}^{2} .$ In particular the dissipation is finite on every positive-time interval.

2.1F1F2F4step 1.2

Signed mass and $L^{1}$ bound. In step 1.2 the right side tends to $0$ when $R \to \infty$ , while $u^{ε} (t) χ_{R} \to u^{ε} (t)$ and $u_{0} χ_{R} \to u_{0}$ pointwise with $∣ u^{ε} (t) ∣ χ_{R} \leq ∣ u^{ε} (t) ∣$ and $∣ u_{0} ∣ χ_{R} \leq ∣ u_{0} ∣$ , and both majorants are integrable by [F1]; dominated convergence gives $\int u^{ε} (t) = \int u_{0}$ , which is signed-mass conservation. For the $L^{1}$ bound let $η_{δ} (s) = \sqrt{s^{2} + δ^{2}} - δ$ , a convex $C^{2}$ function with $0 \leq η_{δ} \leq ∣ \cdot ∣$ , $η_{δ} ↑ ∣ \cdot ∣$ as $δ ↓ 0$ , and let $q_{δ}^{'} = η_{δ}^{'} f^{'}$ with $q_{δ} (0) = 0$ . Testing the balance [F2] with $χ_{R}$ , integrating first on a positive-time interval and passing its lower endpoint to zero by the Lipschitz entropy and $L^{1}$ continuity and using $∣ q_{δ} (s) ∣ \leq L_{M} ∣ s ∣$ and $η_{δ} (s) \leq ∣ s ∣$ on the range of [F1], the cutoff terms vanish in the limit $R \to \infty$ exactly as in step 1.2, and dropping the nonpositive dissipation gives $\int η_{δ} (u^{ε} (t)) \leq \int η_{δ} (u_{0})$ ; monotone convergence in $δ ↓ 0$ yields $∥ u^{ε} (t) ∥_{1} \leq ∥ u_{0} ∥_{1}$ .

3.1F1step 1.3F4∎

Passage to the initial time. The construction gives $u^{ε} \in C ([0, T]; L^{1})$ and $∥ u^{ε} (s) ∥_{\infty} \leq M = ∥ u_{0} ∥_{\infty}$ . Hence $∥ u^{ε} (s) - u_{0} ∥_{2}^{2} \leq 2 M ∥ u^{ε} (s) - u_{0} ∥_{1} ⟶ 0 (s ↓ 0),$ so the endpoint energy in step 1.3 converges to $\frac{1}{2} ∥ u_{0} ∥_{2}^{2}$ . Letting $s ↓ 0$ , monotone convergence for the nonnegative space-time dissipation gives the identity in (iii) for every $t > 0$ ; at $t = 0$ it is immediate. Dropping the nonnegative final energy yields the stated uniform bound for $0 < ε \leq 1$ .

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Viscous solutions contract spatial translates in L-one

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $n \geq 1$ , $0 < ε \leq 1$ , let $f \in C^{2} (R; R^{n})$ satisfy $f (0) = 0$ , and let $u_{0} \in C_{c}^{\infty} (R^{n})$ . Let $u^{ε}$ be the mild viscous solution with initial datum $u_{0}$ from The viscous scalar Cauchy problem with smooth data has a global classical solution. Then for every $h \in R^{n}$ and $t \in [0, T]$ , $∥ u^{ε} (\cdot + h, t) - u^{ε} (\cdot, t) ∥_{1} \leq ∥ u_{0} (\cdot + h) - u_{0} (\cdot) ∥_{1} .$ The estimate depends on $f$ only through a bound for $∣ f^{'} ∣$ on the solution range. Thus it is uniform for any family of such fluxes with a common derivative bound on that range and the same initial datum (Absolute value in an ordered field, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < ε \leq 1$ , $f \in C^{2}$ with $f (0) = 0$ , $u_{0} \in C_{c}^{\infty}$ , the viscous solution $u^{ε}$ , a shift $h \in R^{n}$ , and a spatial cutoff family $χ_{R} (x) = χ (x / R)$ with $χ \in C_{c}^{\infty}$ , $0 \leq χ \leq 1$ , $χ = 1$ on $B_{1}$ and $χ_{R} \to 1$ pointwise.

[F1]

The viscous solution is a classical solution with $u^{ε} \in C ([0, T]; C_{b}) \cap C ([0, T]; L^{1}) \cap C^{1, 2} (R^{n} \times (0, T))$ , its range is the initial range, and it is bounded in $L^{1}$ uniformly on $[0, T]$ (The viscous scalar Cauchy problem with smooth data has a global classical solution, Uniform L-infinity, mass and energy bounds for the viscous approximations).

[F2]

Coordinate chain and product rules give the calculus identities for $C^{1, 2}$ functions (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field). For compactly supported smooth tests, spatial integration by parts follows by enclosing the support in a box, applying Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives in each coordinate with the others fixed, identifying the continuous slice integrals with Lebesgue integrals (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral), and using Fubini's theorem for L^1 functions on a sigma-finite product; boundary terms vanish. Applying this twice transfers the Laplacian to the test function.

[F3]

Dominated convergence on compact sets and in the cutoff limit, and continuity in $C ([0, T]; L^{1})$ permitting the limits $R \to \infty$ and $s ↓ 0$ (Dominated convergence, The space $L^{p} (μ)$ as the quotient by null functions). The smooth cutoffs with $∣ D χ_{R} ∣ \leq C / R$ and $∣ Δ χ_{R} ∣ \leq C / R^{2}$ are supplied by Explicit compactly supported smooth cutoffs.

Proof

technique · direct

1.1F1F2

The translate difference solves a linear equation. Fix $h$ and put $w (t, x) = u^{ε} (t, x + h) - u^{ε} (t, x)$ , so that $w \in C ([0, T]; L^{1}) \cap C^{1, 2}$ by [F1]. Define $a (t, x) = \int_{0}^{1} f^{'} (s u^{ε} (t, x + h) + (1 - s) u^{ε} (t, x)) d s$ ; then $a$ is $C^{1}$ and bounded by $L_{M} = \sup_{∣ s ∣ \leq ∥ u_{0} ∥_{\infty}} ∣ f^{'} (s) ∣$ , and $f (u^{ε} (t, x + h)) - f (u^{ε} (t, x)) = a (t, x) w (t, x)$ . Subtracting the two pointwise viscous equations and using the chain rule gives $w_{t} + div (a w) = ε Δ w$ .

2.1F2step 1.1

The modulus balance. For $δ > 0$ let $η_{δ} (r) = \sqrt{r^{2} + δ^{2}}$ , so that $η_{δ} \in C^{\infty}$ , $η_{δ} \geq ∣ r ∣$ , $∣ η_{δ}^{'} ∣ \leq 1$ , and $η_{δ} (r) - r η_{δ}^{'} (r) = δ^{2} / η_{δ} (r)$ , $η_{δ}^{''} (r) = δ^{2} / η_{δ} (r)^{3}$ . On compact subsets of $R^{n} \times (0, T)$ , using step 1.1 and [F2], $\partial_{t} η_{δ} (w) + div (a η_{δ} (w)) - ε Δ η_{δ} (w) = η_{δ}^{'} (w) (w_{t} + a \cdot \nabla w - ε Δ w) + (div a) η_{δ} (w) - ε η_{δ}^{''} (w) ∣ \nabla w ∣^{2} = \frac{δ^{2}}{η_{δ} (w)} div a - ε \frac{δ^{2} ∣ \nabla w ∣^{2}}{η_{δ} (w)^{3}}$ .

3.1F1F3step 2.1

The limit $δ ↓ 0$ . The second term of step 2.1 is nonpositive, and the first is bounded in absolute value by $δ ∣ div a ∣$ , which tends to $0$ in $L_{l o c}^{1}$ as $δ ↓ 0$ because $div a$ is bounded on compact sets; since $η_{δ} (w) \to ∣ w ∣$ pointwise and $∣ η_{δ} (w) ∣ \leq ∣ w ∣ + δ$ , dominated convergence gives the distributional inequality $\partial_{t} ∣ w ∣ + div (a ∣ w ∣) \leq ε Δ ∣ w ∣$ on $Π_{T}$ .

4.1F1F2F3step 3.1

Cutoff estimate. Test step 3.1 with $χ_{R} (x)$ times a nonnegative smooth time test. In distributions in time this gives $\frac{d}{d t} \int ∣ w (t) ∣ χ_{R} \leq C (L_{M} R^{- 1} + ε R^{- 2}) ∥ w (t) ∥_{1}$ . Approximating the indicator of $(s, t)$ by smooth time cutoffs and using the $L^{1}$ continuity of $w$ gives, for all $0 < s < t < T$ , $\int ∣ w (t) ∣ χ_{R} \leq \int ∣ w (s) ∣ χ_{R} + C (L_{M} R^{- 1} + ε R^{- 2}) \int_{s}^{t} ∥ w (r) ∥_{1} d r$ . The time integral is finite by [F1]. Dominated convergence as $R \to \infty$ yields $∥ w (t) ∥_{1} \leq ∥ w (s) ∥_{1}$ ; continuity includes $t = T$ .

5.1step 4.1F1F3∎

Conclusion. Letting $s ↓ 0$ in step 4.1 and using the $C ([0, T]; L^{1})$ continuity of $w$ and $w (0, \cdot) = u_{0} (\cdot + h) - u_{0} (\cdot)$ gives $∥ u^{ε} (\cdot + h, t) - u^{ε} (\cdot, t) ∥_{1} = ∥ w (t) ∥_{1} \leq ∥ w (0) ∥_{1} = ∥ u_{0} (\cdot + h) - u_{0} (\cdot) ∥_{1}$ for every $t \in [0, T]$ . Every constant used depends on $f$ only through $L_{M}$ , the bound for $∣ f^{'} ∣$ on the solution range, so the estimate is uniform over such flux families.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The Kruzhkov doubling inequality for two entropy solutions

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ , and let $u, v$ be bounded Kruzhkov entropy solutions on $Π_{T}$ in the sense of Kruzhkov entropy solutions. Then, in the sense of distributions on $Π_{T}$ , $\partial_{t} ∣ u - v ∣ + {div}_{x} (sgn (u - v) (f (u) - f (v))) \leq 0.$ Equivalently, for every nonnegative $φ \in C_{c}^{\infty} (Π_{T})$ , $\int_{Π_{T}} (∣ u - v ∣ φ_{t} + sgn (u - v) (f (u) - f (v)) \cdot \nabla_{x} φ) d x d t \geq 0.$ Here $sgn (0) = 0$ . Together with the weak equation this is the doubling-variables inequality from which uniqueness and the local $L^{1}$ contraction are read off (Distribution, Distributional derivative).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , $f \in C^{1}$ , bounded entropy solutions $u, v$ on $Π_{T}$ , a nonnegative test function $φ \in C_{c}^{\infty} (Π_{T})$ , and nonnegative unit-mass even mollifiers $θ_{h}$ on $R$ and $ρ_{h}$ on $R^{n}$ with $\int ρ_{h} = 1$ , $supp θ_{h} \subset (- h, h)$ and $supp ρ_{h} \subset B_{h}$ (A radial mollifier family in Rn, Convolution of a distribution with a test function).

[F1]

For every $k \in R$ the pair $(η_{k}, q_{k})$ with $η_{k} (s) = ∣ s - k ∣$ , $q_{k} (s) = sgn (s - k) (f (s) - f (k))$ is a convex entropy pair with $q_{k}^{'} = η_{k}^{'} f^{'}$ , and each of $u, v$ satisfies the corresponding distributional inequality against every nonnegative test function (Kruzhkov entropy solutions, Convex entropy--entropy flux pairs, Absolute value in an ordered field).

[F2]

Fubini and dominated convergence apply on compact supports (Fubini's theorem for L^1 functions on a sigma-finite product, Dominated convergence). After multiplication by a fixed cutoff, $u, v$ lie in $L^{1} (R^{n + 1})$ and their translations are norm continuous under Countable Choice ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ , The Axiom of Countable Choice ( ${A C}_{ω}$ ), The space $L^{p} (μ)$ as the quotient by null functions). This is the $L^{1}$ diagonal interface; distributional mollifier convergence alone would not supply it.

Proof

technique · direct

1.1F2given

The doubled test function. For $h > 0$ smaller than half the distance of the temporal support of $φ$ from ${0, T}$ , set $g_{h} (t, x, τ, y) = φ (\frac{t + τ}{2}, \frac{x + y}{2}) θ_{h} (t - τ) ρ_{h} (x - y)$ ; it is nonnegative and smooth with compact support in each pair of variables, and $\partial_{t} g_{h} + \partial_{τ} g_{h} = (\partial_{t} φ) (\frac{t + τ}{2}, \frac{x + y}{2}) θ_{h} (t - τ) ρ_{h} (x - y)$ , $\nabla_{x} g_{h} + \nabla_{y} g_{h} = (\nabla φ) (\frac{t + τ}{2}, \frac{x + y}{2}) θ_{h} (t - τ) ρ_{h} (x - y)$ .

1.2F2given

The inequality for $u$ with state $k = v (τ, y)$ . For almost every $(τ, y)$ the constant $k = v (τ, y)$ is admissible in [F1], and testing the entropy inequality for $u$ by the nonnegative function $g_{h} (\cdot, \cdot, τ, y)$ gives $\int_{Π_{T}} (∣ u (t, x) - v (τ, y) ∣ \partial_{t} g_{h} + sgn (u - v (τ, y)) (f (u) - f (v (τ, y))) \cdot \nabla_{x} g_{h}) d x d t \geq 0$ ; the integrand is bounded by a constant times the compactly supported smooth $g_{h}$ and its derivatives, so the left side is a bounded measurable function of $(τ, y)$ .

2.1F2step 1.2

Adding the symmetric inequality. Integrating the inequality of step 1.2 over $(τ, y) \in Π_{T}$ , and likewise testing the entropy inequality for $v$ with $k = u (t, x)$ and integrating over $(t, x)$ , then adding, Fubini's theorem gives $0 \leq \int ⁣ ⁣ \int_{Π_{T} \times Π_{T}} [∣ u - v ∣ (\partial_{t} g_{h} + \partial_{τ} g_{h}) + sgn (u - v) (f (u) - f (v)) \cdot (\nabla_{x} g_{h} + \nabla_{y} g_{h})]$ ; the two integrals have the same bounded integrand because of the symmetry of $g_{h}$ .

3.1F2step 1.1step 2.1∎

Passing to the diagonal. Put $z = (t, x)$ and $z^{'} = (τ, y) = z - h^{'}$ . On a fixed compact set containing the doubled supports, translation continuity gives $∥ v (\cdot - h^{'}) - v ∥_{1} \to 0$ uniformly for $∣ h^{'} ∣ \leq 2 h$ . The map $Q (a, b) = sgn (a - b) (f (a) - f (b))$ is Lipschitz in each variable on the common bounded range: when a variable crosses the other one, split the interval at that point and use $Q (b, b) = 0$ and the flux Lipschitz bound. Hence replacing $v (z^{'})$ by $v (z)$ changes the doubled integral by at most $C \sup_{∣ h^{'} ∣ \leq 2 h} ∥ v (\cdot - h^{'}) - v ∥_{1}$ . Replacing $D φ ((z + z^{'}) / 2)$ by $D φ (z)$ has error $O (h)$ by smoothness, boundedness and unit kernel mass. Step 2.1 therefore converges to $\int_{Π_{T}} (∣ u - v ∣ φ_{t} + Q (u, v) \cdot \nabla φ) \geq 0$ .

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Local $L^{1}$ contraction for two entropy solutions

Statement

Assume Countable Choice. Let $n \geq 1$ , $T > 0$ , and $f ⁣ : R \to R^{n}$ be $C^{1}$ . Let $M, L \geq 0$ and assume $∣ f (a) - f (b) ∣ \leq L ∣ a - b ∣$ for all $a, b \in [- M, M]$ . Let $u, v$ be bounded Kruzhkov entropy solutions on $Π_{T}$ in the sense of Kruzhkov entropy solutions, with $∣ u ∣, ∣ v ∣ \leq M$ almost everywhere and initial data $u_{0}, v_{0} \in L^{\infty} \cap L_{l o c}^{1}$ . For each fixed $x_{0} \in R^{n}$ and $R > 0$ , for almost every $t \in (0, T)$ satisfying $L t < R$ , $\int_{B (x_{0}, R - L t)} ∣ u (t, x) - v (t, x) ∣ d x \leq \int_{B (x_{0}, R)} ∣ u_{0} (x) - v_{0} (x) ∣ d x .$ In particular, for each fixed $x_{0}, R$ , if $u_{0} = v_{0}$ almost everywhere on $B (x_{0}, R)$ , then $u (t, \cdot) = v (t, \cdot)$ almost everywhere on $B (x_{0}, R - L t)$ for almost every $t$ with $L t < R$ . The exceptional null set may depend on $x_{0}$ and $R$ .

Facts & Assumptions

Given: $n \geq 1$ , $T > 0$ , $f \in C^{1} (R; R^{n})$ , constants $M, L \geq 0$ with $∣ f (a) - f (b) ∣ \leq L ∣ a - b ∣$ on $[- M, M]$ , bounded Kruzhkov entropy solutions $u, v$ on $Π_{T}$ with $∣ u ∣, ∣ v ∣ \leq M$ almost everywhere and initial data $u_{0}, v_{0} \in L^{\infty} \cap L_{l o c}^{1} (R^{n})$ , a centre $x_{0} \in R^{n}$ and radius $R > 0$ , and the abbreviations $w = ∣ u - v ∣$ , $q = sgn (u - v) (f (u) - f (v))$ .

[F1]

Kato's inequality: for every nonnegative $φ \in C_{c}^{\infty} (Π_{T})$ , $\int_{Π_{T}} (w φ_{t} + q \cdot \nabla_{x} φ) d x d t \geq 0$ . Since $∣ u ∣, ∣ v ∣ \leq M$ almost everywhere and $f$ is $L$ -Lipschitz on $[- M, M]$ , also $∣ q ∣ \leq L w$ almost everywhere (The Kruzhkov doubling inequality for two entropy solutions, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction, Kruzhkov entropy solutions).

[F2]

Strong local $L^{1}$ initial traces: for every compact $K \subseteq R^{n}$ , $\lim_{δ ↓ 0} {ess sup}_{0 < t < δ} \int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x = 0$ , and the same holds for $v$ and $v_{0}$ (Kruzhkov entropy solutions).

[F3]

Cutoff profiles: for every $σ > 0$ there is a smooth nonincreasing $β_{σ} ⁣ : R \to [0, 1]$ with $β_{σ} = 1$ on $(- \infty, R - σ]$ and $β_{σ} = 0$ on $[R, \infty)$ , obtained by integrating a nonnegative smooth bump supported in $(R - σ, R)$ ; then $β_{σ}^{'} \leq 0$ and $Φ_{σ} (t, x) = β_{σ} (∣ x - x_{0} ∣ + L t)$ satisfies $\partial_{t} Φ_{σ} + L ∣ \nabla_{x} Φ_{σ} ∣ = 0$ on the region where $∣ x - x_{0} ∣ + L t > R - σ$ , and $Φ_{σ} = 1$ on the region where $∣ x - x_{0} ∣ + L t \leq R - σ$ ; hence $Φ_{σ}$ is smooth on the slab $0 < t < t_{3}$ whenever $L t_{3} < R - σ$ , has compact spatial support contained in $B (x_{0}, R)$ , and vanishes identically for $t \geq R / L$ if $L > 0$ (A Euclidean bump for a compact set inside an open set, Open ball, closed ball and sphere in a metric space).

[F4]

Slice functions: $F_{σ} (t) = \int_{R^{n}} w (t, x) Φ_{σ} (t, x) d x$ is well defined for almost every $t$ and locally integrable on its interval of definition, because $w$ is bounded and $Φ_{σ}$ is bounded with compact spatial support; hence almost every point is a Lebesgue point of $F_{σ}$ , and the intersection of countably many full-measure sets is again full measure. Dominated and monotone convergence justify limits of integrals with uniformly bounded integrands against fixed integrable functions (Lebesgue differentiation theorem on $R^{n}$ , Dominated convergence, The space $L^{p} (μ)$ as the quotient by null functions, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F3F4

Cutoff inequalities on a time slab. Fix $t_{3} \in (0, T)$ with $L t_{3} < R$ — for $L > 0$ such $t_{3}$ exist by taking $t_{3} < \min {T, R / L}$ , and for $L = 0$ every $t_{3} \in (0, T)$ works — and fix $σ \in (0, R - L t_{3})$ . Let $Φ_{σ}$ be as in [F3] and set $F_{σ} (t) = \int_{R^{n}} w (t, x) Φ_{σ} (t, x) d x$ and $G_{σ} (t) = \int_{R^{n}} (w \partial_{t} Φ_{σ} + q \cdot \nabla_{x} Φ_{σ}) (t, x) d x$ for $t \in (0, t_{3})$ . Because $β_{σ}^{'} \leq 0$ and [F3] holds, $G_{σ} \leq \int_{R^{n}} w (\partial_{t} Φ_{σ} + L ∣ \nabla_{x} Φ_{σ} ∣) d x = 0$ by [F1]. For nonnegative $η \in C_{c}^{\infty} ((0, t_{3}))$ the function $φ = η Φ_{σ}$ is an admissible nonnegative test function in [F1], since $Φ_{σ}$ is smooth on the slab and compactly supported in $x$ ; hence $0 \leq \int_{0}^{t_{3}} (F_{σ} η^{'} + G_{σ} η) \leq \int_{0}^{t_{3}} F_{σ} η^{'}$ , that is, $\int_{0}^{t_{3}} F_{σ} η^{'} \geq 0$ .

2.1F4step 1.1

Monotonicity in time. Fix a nonnegative smooth bump $ζ$ supported in $(0, 1)$ with $\int_{0}^{1} ζ = 1$ and put $H (r) = \int_{- \infty}^{r} ζ$ . For $0 < s < t < t_{3}$ and small $ε > 0$ , the function $η_{ε} (τ) = H (\frac{τ - s}{ε}) - H (\frac{τ - t}{ε})$ is admissible in step 1.1 and $η_{ε}^{'} \to δ_{s} - δ_{t}$ as $ε ↓ 0$ . At Lebesgue points $s < t$ of $F_{σ}$ , step 1.1 gives $F_{σ} (s) - F_{σ} (t) = \lim_{ε ↓ 0} \int_{0}^{t_{3}} F_{σ} η_{ε}^{'} \geq 0$ , so $F_{σ} (s) \geq F_{σ} (t)$ for all Lebesgue points $0 < s < t < t_{3}$ of $F_{σ}$ , a full-measure set of pairs by [F4].

3.1F2F4step 2.1

The limit as $s ↓ 0$ . We claim ${ess lim}_{s ↓ 0} F_{σ} (s) = \int_{R^{n}} ∣ u_{0} (x) - v_{0} (x) ∣ β_{σ} (∣ x - x_{0} ∣) d x$ . Indeed, the difference is bounded by $\int_{B (x_{0}, R)} ∣ u (s) - u_{0} ∣ β_{σ} + \int_{B (x_{0}, R)} ∣ v (s) - v_{0} ∣ β_{σ} + \int_{R^{n}} ∣ u_{0} - v_{0} ∣ ∣ β_{σ} (∣ x - x_{0} ∣ + L s) - β_{σ} (∣ x - x_{0} ∣) ∣$ ; the first two terms tend to $0$ by [F2], since $β_{σ}$ is supported in $B (x_{0}, R)$ , and the third tends to $0$ because $β_{σ}$ has bounded derivative and $∣ u_{0} - v_{0} ∣ \in L_{l o c}^{1}$ . Combining with step 2.1 and letting $s ↓ 0$ through Lebesgue points of $F_{σ}$ , for almost every $t \in (0, t_{3})$ , $F_{σ} (t) \leq \int_{R^{n}} ∣ u_{0} - v_{0} ∣ β_{σ} (∣ x - x_{0} ∣) d x$ .

4.1F4step 3.1∎

Removing the cutoff. Let $σ_{j} ↓ 0$ with $σ_{j} < R - L t_{3}$ and intersect the full-measure sets of step 3.1 over all $j$ using [F4]: for almost every $t \in (0, t_{3})$ the inequality of step 3.1 with $σ = σ_{j}$ holds for every $j$ . For such $t$ , $β_{σ_{j}} (∣ x - x_{0} ∣ + L t) \to 1_{{∣ x - x_{0} ∣ < R - L t}} (x)$ pointwise away from the sphere $∣ x - x_{0} ∣ = R - L t$ , and the corresponding integrands are dominated by $w (t, \cdot) 1_{B (x_{0}, R)}$ respectively $∣ u_{0} - v_{0} ∣ 1_{B (x_{0}, R)}$ , which are integrable; hence dominated convergence gives $\int_{B (x_{0}, R - L t)} ∣ u - v ∣ (t, x) d x \leq \int_{B (x_{0}, R)} ∣ u_{0} - v_{0} ∣ (x) d x$ . Every $t \in (0, T)$ with $L t < R$ lies in $(0, t_{3})$ for some admissible $t_{3}$ — put $H = \min {T, R / L}$ for $L > 0$ , and $H = T$ for $L = 0$ , and use the explicit sequence $t_{3}^{(j)} = H (1 - 1 / (j + 1)) ↑ H$ — so the estimate holds for almost every such $t$ , with exceptional set depending on $x_{0}, R$ . If $u_{0} = v_{0}$ almost everywhere on $B (x_{0}, R)$ the right-hand side vanishes, so $u (t, \cdot) = v (t, \cdot)$ almost everywhere on $B (x_{0}, R - L t)$ for almost every such $t$ .

Remarks

The global $L^{1}$ estimate is Global $L^{1}$ contraction from the local estimate (Open ball, closed ball and sphere in a metric space).

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Uniqueness, comparison and order preservation of entropy solutions

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $n \geq 1$ , $T > 0$ and let $f ⁣ : R \to R^{n}$ be $C^{1}$ (hence Lipschitz on bounded intervals).

(i) If $u, v$ are bounded Kruzhkov entropy solutions on $Π_{T}$ in the sense of Kruzhkov entropy solutions with $∣ u ∣, ∣ v ∣ \leq M$ and $u_{0} \leq v_{0}$ almost everywhere, then $u \leq v$ almost everywhere on $Π_{T}$ .

(ii) There is at most one bounded Kruzhkov entropy solution with a given initial datum $u_{0} \in L^{\infty} \cap L_{l o c}^{1}$ ; if $u_{0} = v_{0}$ almost everywhere on the whole of $R^{n}$ , then $u = v$ almost everywhere on $Π_{T}$ .

(iii) The positive part contracts: $\int_{R^{n}} (u (t, \cdot) - v (t, \cdot))_{+} d x \leq \int_{R^{n}} (u_{0} - v_{0})_{+} d x$ for almost every $t \in (0, T)$ whenever both sides are finite (Absolute value in an ordered field, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , a $C^{1}$ flux $f$ , bounded Kruzhkov entropy solutions $u, v$ on $Π_{T}$ with common essential bound $M$ , and a constant $L \geq 0$ with $∣ f (a) - f (b) ∣ \leq L ∣ a - b ∣$ for $a, b \in [- M, M]$ .

[F1]

Entropy solutions are distributional weak solutions: $\partial_{t} (u - v) + {div}_{x} (f (u) - f (v)) = 0$ in $D^{'} (Π_{T})$ ; moreover each of $u, v$ has the strong local $L^{1}$ initial trace: for every compact $K \subseteq R^{n}$ , ${ess sup}_{0 < t < δ} \int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x \to 0$ as $δ ↓ 0$ , and likewise for $v$ (Kruzhkov entropy solutions).

[F2]

Kato's inequality: $\partial_{t} ∣ u - v ∣ + {div}_{x} (sgn (u - v) (f (u) - f (v))) \leq 0$ in $D^{'} (Π_{T})$ (The Kruzhkov doubling inequality for two entropy solutions).

[F3]

Positive-part identities: for every $r \in R$ , $(r)_{+} = \frac{1}{2} (∣ r ∣ + r)$ and $1_{{r > 0}} = \frac{1}{2} (sgn (r) + 1)$ for $r \neq 0$ ; at $u = v$ the flux difference is zero, so the flux identity remains valid with $sgn (0) = 0$ , so adding [F1] and [F2] gives $\partial_{t} (u - v)_{+} + {div}_{x} (1_{{u > v}} (f (u) - f (v))) \leq 0$ in $D^{'} (Π_{T})$ ; writing $w = (u - v)_{+}$ and $q = 1_{{u > v}} (f (u) - f (v))$ , the Lipschitz hypothesis gives $∣ q ∣ \leq L w$ almost everywhere, since $∣ u ∣, ∣ v ∣ \leq M$ and $f$ is $L$ -Lipschitz on $[- M, M]$ (Absolute value in an ordered field).

[F4]

Cutoff machinery of Local $L^{1}$ contraction for two entropy solutions: for $t_{3} \in (0, T)$ with $L t_{3} < R$ and $0 < σ < R - L t_{3}$ there is a smooth nonincreasing $β_{σ}$ with $β_{σ} = 1$ on $(- \infty, R - σ]$ and $β_{σ} = 0$ on $[R, \infty)$ , and $Φ_{σ} (t, x) = β_{σ} (∣ x - x_{0} ∣ + L t)$ satisfies $\partial_{t} Φ_{σ} + L ∣ \nabla_{x} Φ_{σ} ∣ = 0$ , is compactly supported in $x$ , and is admissible as a test factor on $(0, t_{3})$ ; for $w \geq 0$ with strong local $L^{1}$ initial trace $w_{0}$ and $∣ q ∣ \leq L w$ , testing $\partial_{t} w + {div}_{x} q \leq 0$ against $η Φ_{σ}$ and letting $σ ↓ 0$ along a decreasing sequence yields $\int_{B (x_{0}, R - L t)} w (t, x) d x \leq \int_{B (x_{0}, R)} w_{0} (x) d x$ for almost every $t \in (0, T)$ with $L t < R$ ; the argument uses Lebesgue points of $t \mapsto \int w Φ_{σ}$ , monotone and dominated convergence (Dominated convergence, Monotone convergence for the integral, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F1F2F3

The positive-part inequality. By [F1] and [F2], the sum of the weak equation for $(u - v)$ and Kato's inequality is the distributional inequality $\partial_{t} w + {div}_{x} q \leq 0$ with $w = (u - v)_{+}$ and $q = 1_{{u > v}} (f (u) - f (v))$ by the identities of [F3], and $∣ q ∣ \leq L w$ almost everywhere.

2.1F4step 1.1

Local positive-part estimate. Apply the cutoff computation [F4] to the pair $(w, q)$ of step 1.1 with any centre $x_{0}$ and radius $R$ : $\int_{B (x_{0}, R - L t)} (u - v)_{+} (t, x) d x \leq \int_{B (x_{0}, R)} (u_{0} - v_{0})_{+} (x) d x$ for almost every $t \in (0, T)$ with $L t < R$ . Indeed the structural hypotheses of [F4] are met: the strong local $L^{1}$ trace of $w$ at $0$ is $w_{0} = (u_{0} - v_{0})_{+}$ because $∣ w (t, x) - w_{0} (x) ∣ \leq ∣ u (t, x) - u_{0} (x) ∣ + ∣ v (t, x) - v_{0} (x) ∣$ almost everywhere, the positive part being $1$ -Lipschitz, and $∣ q ∣ \leq L w$ holds by step 1.1; the cutoff, Lebesgue-point, initial-trace and $σ ↓ 0$ steps are those of the proof of Local $L^{1}$ contraction for two entropy solutions with $∣ u - v ∣$ replaced by $(u - v)_{+}$ .

3.1F4step 2.1

Order preservation. Assume $u_{0} \leq v_{0}$ almost everywhere, so $(u_{0} - v_{0})_{+} = 0$ almost everywhere and the right-hand side of step 2.1 vanishes for every centre and radius. Take centres $x_{0} = 0$ and radii $R_{m} = L T + m$ , $m \geq 1$ , so that $B (0, R_{m} - L t) \supseteq B (0, m - L T)$ for every $t \in (0, T)$ ; intersecting the countably many full-measure sets of times supplied by step 2.1, for almost every $t \in (0, T)$ one has $\int_{B (0, m - L T)} (u - v)_{+} (t, x) d x = 0$ for every $m$ with $m > L T$ , hence $(u - v)_{+} (t, \cdot) = 0$ almost everywhere on the union $⋃_{m} B (0, m - L T) = R^{n}$ . By Fubini, $(u - v)_{+} = 0$ almost everywhere on $Π_{T}$ , that is, $u \leq v$ almost everywhere.

4.1step 3.1

Uniqueness. If $u, v$ are bounded entropy solutions with the same datum $u_{0} = v_{0}$ , then both $u_{0} \leq v_{0}$ and $v_{0} \leq u_{0}$ hold almost everywhere, so step 3.1 gives $u \leq v$ and $v \leq u$ almost everywhere on $Π_{T}$ , whence $u = v$ almost everywhere. This proves both assertions of (ii).

5.1F4step 2.1∎

Positive-part contraction. Let $u, v$ be any two bounded entropy solutions with both integrals finite; step 2.1 with centre $0$ and radii $R_{m} = L T + m$ gives $\int_{B (0, R_{m} - L t)} (u - v)_{+} \leq \int_{B (0, R_{m})} (u_{0} - v_{0})_{+} \leq \int_{R^{n}} (u_{0} - v_{0})_{+} < \infty$ for almost every $t$ outside a null set $N_{m}$ . On the complement of the null set $⋃_{m} N_{m}$ , all these inequalities hold, and monotone convergence over the increasing balls $B (0, R_{m} - L t) ↑ R^{n}$ gives $\int_{R^{n}} (u (t, \cdot) - v (t, \cdot))_{+} d x \leq \int_{R^{n}} (u_{0} - v_{0})_{+} d x$ , which is (iii).

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Finite propagation for scalar conservation laws

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $n \geq 1$ , $T > 0$ , and let $f ⁣ : R \to R^{n}$ be $C^{1}$ (hence Lipschitz on bounded intervals), with Lipschitz constant $L \geq 0$ on the common essential range of the solutions below. Let $u, v$ be bounded Kruzhkov entropy solutions on $Π_{T}$ in the sense of Kruzhkov entropy solutions. Fix $x_{0} \in R^{n}$ and $R > 0$ . If $u_{0} = v_{0}$ almost everywhere on $B (x_{0}, R)$ , then for almost every $t \in (0, T)$ with $L t < R$ , $u (t, \cdot) = v (t, \cdot) almost everywhere on B (x_{0}, R - L t) .$ For $L = 0$ this holds for almost every $t \in (0, T)$ on the stationary ball $B (x_{0}, R)$ . In particular, if $u_{0} = 0$ almost everywhere outside $B (x_{0}, R)$ , then $u (t, x) = 0$ for almost every $(t, x) \in Π_{T}$ with $∣ x - x_{0} ∣ > R + L t$ . The conclusions are almost-everywhere statements for each fixed cone; an every-time claim requires a chosen time-continuous representative (Open ball, closed ball and sphere in a metric space).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , $f \in C^{1}$ , bounded Kruzhkov entropy solutions $u, v$ on $Π_{T}$ with common essential bound $M$ and $∣ u ∣, ∣ v ∣ \leq M$ almost everywhere, a constant $L \geq 0$ with $∣ f (a) - f (b) ∣ \leq L ∣ a - b ∣$ for $a, b \in [- M, M]$ , a centre $x_{0} \in R^{n}$ , and $R > 0$ .

[F1]

Local $L^{1}$ contraction: for almost every $t \in (0, T)$ with $L t < R$ , $\int_{B (x_{0}, R - L t)} ∣ u (t, x) - v (t, x) ∣ d x \leq \int_{B (x_{0}, R)} ∣ u_{0} (x) - v_{0} (x) ∣ d x$ , with an exceptional null set depending on $x_{0}, R$ ; when $L = 0$ the time condition is vacuous and the ball is stationary (Local $L^{1}$ contraction for two entropy solutions, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction, Kruzhkov entropy solutions).

[F2]

The function identically $0$ on $Π_{T}$ is a Kruzhkov entropy solution with initial datum $0$ for every flux $f$ : for each $k \in R$ the functions $η_{k} (0) = ∣ k ∣$ and $q_{k} (0) = sgn (- k) (f (0) - f (k))$ are constant, so $\partial_{t} η_{k} (0) + {div}_{x} q_{k} (0) = 0$ in distributions, and the strong local $L^{1}$ initial condition holds because $\int_{K} ∣ 0 - 0 ∣ d x = 0$ for every compact $K$ (Kruzhkov entropy solutions).

[F3]

A nonnegative function in $L_{l o c}^{1}$ has vanishing integral over an open ball if and only if it vanishes almost everywhere there; almost-everywhere statements are statements about equivalence classes (The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

Countable measure bookkeeping: $Q$ is countable and dense in $R$ ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable), hence finite products are countable (A product of two at most countable sets is at most countable). To approximate a point of $R^{n}$ within $δ$ , approximate each coordinate by a rational within $δ / \sqrt{n}$ ; this proves density of $Q^{n}$ . a countable intersection of full-measure sets of times is full measure, and a countable union of null subsets of $Π_{T}$ is null; Fubini gives the section-to-product nullity implication (Fubini's theorem for L^1 functions on a sigma-finite product). A countable union of null sets is null: finite-union indicators are bounded by the finite sums of the null-set indicators, and monotone convergence passes to their increasing union. Limits of integrals over expanding balls are covered by monotone and dominated convergence (Monotone convergence for the integral, Dominated convergence, Open ball, closed ball and sphere in a metric space).

Proof

technique · direct

1.1F1F3

The local estimate with vanishing right-hand side. Fix $x_{0}, R$ and suppose first that $u_{0} = v_{0}$ almost everywhere on $B (x_{0}, R)$ . By [F1], for almost every $t \in (0, T)$ with $L t < R$ , $\int_{B (x_{0}, R - L t)} ∣ u (t, x) - v (t, x) ∣ d x \leq \int_{B (x_{0}, R)} ∣ u_{0} (x) - v_{0} (x) ∣ d x = 0,$ so $u (t, \cdot) = v (t, \cdot)$ almost everywhere on $B (x_{0}, R - L t)$ by [F3]. If $L = 0$ the condition $L t < R$ reads $0 < R$ and is automatic, the ball $B (x_{0}, R - L t) = B (x_{0}, R)$ is stationary, and the statement holds for almost every $t \in (0, T)$ .

2.1F2step 1.1

The support claim for the atomic ball. Suppose now that $u_{0} = 0$ almost everywhere outside $B (x_{0}, R)$ and let $q \in R^{n}$ , $r_{0} > 0$ with $B (q, r_{0}) \subseteq R^{n} ∖ \overline{B} (x_{0}, R)$ . Then $u_{0} = 0$ almost everywhere on $B (q, r_{0})$ , so step 1.1 applied to the pair $(u, 0)$ with centre $q$ and radius $r_{0}$ — legitimate because $0$ is an entropy solution with datum $0$ by [F2] — gives $u (t, \cdot) = 0$ almost everywhere on $B (q, r_{0} - L t)$ for almost every $t$ with $L t < r_{0}$ .

3.1F4step 2.1

Covering the exterior cone. Let $Q$ consist of rational pairs $(q, r_{0}) \in Q^{n} \times Q_{> 0}$ satisfying $r_{0} < ∣ q - x_{0} ∣ - R$ , and let $C (q, r_{0}) = {(t, x) : L t < r_{0}, ∣ x - q ∣ < r_{0} - L t}$ . If $∣ x - x_{0} ∣ > R + L t$ , choose rational $q$ sufficiently close to $x$ that $L t + ∣ x - q ∣ < ∣ q - x_{0} ∣ - R$ , then a rational $r_{0}$ between these bounds. Thus $B (q, r_{0})$ lies in the strict exterior of the initial ball and $(t, x) \in C (q, r_{0})$ . The countable family of these cones covers the strict exterior cone.

4.1F4step 2.1step 3.1∎

Conclusion of the support claim. For each $(q, r_{0}) \in Q$ , step 2.1 exhibits a null set of times $t$ with $L t < r_{0}$ such that $u (t, \cdot) \neq 0$ on a positive-measure subset of $B (q, r_{0} - L t)$ ; hence the set $N (q, r_{0}) = {(t, x) \in C (q, r_{0}) ⁣ : u (t, x) \neq 0}$ is a null subset of $Π_{T}$ , by Fubini: its sections are null for almost every time, and bounded spatial sections at the exceptional null set of times contribute zero. A countable union of null sets is null by [F4], so $N = ⋃_{(q, r_{0}) \in Q} N (q, r_{0})$ is null, and by step 3.1 the set ${(t, x) \in Π_{T} ⁣ : ∣ x - x_{0} ∣ > R + L t, u (t, x) \neq 0}$ is contained in $N$ . Therefore $u = 0$ almost everywhere in the exterior cone.

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Vanishing-viscosity families are locally precompact in $L^{1}$

Statement

Assume Dependent Choice. Let $n \geq 1$ , $T > 0$ , $u_{0} \in C_{c}^{\infty} (R^{n})$ , and put $M = ∥ u_{0} ∥_{\infty}$ . Let $(f_{j})_{j \geq 1}$ be $C^{2}$ fluxes $f_{j} ⁣ : R \to R^{n}$ with $f_{j} (0) = 0$ and $\sup_{j} \sup_{∣ s ∣ \leq M + 1} ∣ f_{j}^{'} (s) ∣ < \infty .$ For $0 < ε_{j} \leq 1$ with $ε_{j} ↓ 0$ , let $u^{j}$ be the global mild classical viscous solution with flux $f_{j}$ and datum $u_{0}$ , as supplied by The viscous scalar Cauchy problem with smooth data has a global classical solution. Then every subsequence has a further subsequence converging in $L^{1} (K)$ for every compact $K ⋐ Π_{T}$ to $u \in L^{\infty} \cap L_{l o c}^{1} (Π_{T})$ with $∥ u ∥_{\infty} \leq M$ ; a further subsequence converges almost everywhere on $Π_{T}$ . The limit has a representative in $C ([0, T]; L_{l o c}^{1} (R^{n}))$ with $u (0) = u_{0}$ in $L_{l o c}^{1}$ . In particular this applies to a $C^{1}$ -convergent smooth approximation of one flux. The extraction uses Dependent Choice; energy dissipation alone does not give this strong compactness (Open ball, closed ball and sphere in a metric space, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Dependent Choice, $n \geq 1$ , $T > 0$ , $u_{0} \in C_{c}^{\infty} (R^{n})$ , $M = ∥ u_{0} ∥_{\infty}$ , $C^{2}$ fluxes $f_{j}$ with $f_{j} (0) = 0$ and $L : = \sup_{j} \sup_{∣ s ∣ \leq M + 1} ∣ f_{j}^{'} (s) ∣ < \infty$ , $0 < ε_{j} \leq 1$ with $ε_{j} ↓ 0$ , and the viscous solutions $u^{j}$ of $u_{t}^{j} + {div}_{x} f_{j} (u^{j}) = ε_{j} Δ u^{j}$ on $Π_{T}$ with $u^{j} (0, \cdot) = u_{0}$ .

[F1]

Each $u^{j}$ is a classical global solution with $u^{j} \in C ([0, T]; C_{b} (R^{n})) \cap C ([0, T]; L^{1} (R^{n})) \cap C^{1, 2} (R^{n} \times (0, T))$ , and $u^{j} (t, \cdot) \to u_{0}$ in $L^{1} (R^{n})$ as $t ↓ 0$ (The viscous scalar Cauchy problem with smooth data has a global classical solution).

[F2]

Uniform bounds: $∣ u^{j} (t, x) ∣ \leq M$ and $∥ u^{j} (t, \cdot) ∥_{1} \leq ∥ u_{0} ∥_{1}$ for all $j \geq 1$ and $t \in [0, T]$ (Uniform L-infinity, mass and energy bounds for the viscous approximations).

[F3]

Uniform spatial modulus: for all $j$ , $t \in [0, T]$ and $z \in R^{n}$ , $∥ u^{j} (t, \cdot + z) - u^{j} (t, \cdot) ∥_{1} \leq ∥ u_{0} (\cdot + z) - u_{0} (\cdot) ∥_{1} \leq ω_{0} (∣ z ∣)$ , where $ω_{0} (r) = \sup_{∣ z ∣ \leq r} ∥ u_{0} (\cdot + z) - u_{0} (\cdot) ∥_{1} ↓ 0$ as $r ↓ 0$ by uniform continuity of the compactly supported $u_{0}$ ; the estimate depends on $f_{j}$ only through the derivative bound $L$ on the range, so it is uniform in $j$ (Viscous solutions contract spatial translates in L-one).

[F4]

Mollification and cutoffs: for an even mollifier $ϱ_{h}$ with support in $B_{h}$ and a bounded compactly supported $F \in L^{\infty}$ , the convolution $g = ϱ_{h} * F$ is smooth with $\nabla g = (\nabla ϱ_{h}) * F$ , $Δ g = (Δ ϱ_{h}) * F$ (differentiation under the integral sign via difference quotients and dominated convergence) and $∥ D^{α} g ∥_{\infty} \leq C h^{- ∣ α ∣}$ for $∣ α ∣ \leq 2$ ; Fubini's theorem gives $\int g w = \int F (ϱ_{h} * w) d x$ whenever $w \in L_{l o c}^{1}$ and $F$ is bounded with compact support, and the mollification error obeys $∥ ϱ_{h} * w - w ∥_{L^{1} (B_{R + ρ / 2})} \leq \sup_{∣ y ∣ \leq h} ∥ w (\cdot + y) - w (\cdot) ∥_{L^{1} (B_{R + ρ})}$ (A radial mollifier family in Rn, Fubini's theorem for L^1 functions on a sigma-finite product, Dominated convergence).

[F5]

For every $R > 0$ , $ρ \in (0, 1)$ and every compact cylinder there exist smooth cutoffs $0 \leq χ \in C_{c}^{\infty} (B_{R + ρ / 2})$ with $χ = 1$ on $B_{R}$ , and $ψ \in C_{c}^{\infty} (R^{n})$ , $θ \in C_{c}^{\infty} ((0, T))$ equal to $1$ on the spatial and temporal projections of that cylinder (Open ball, closed ball and sphere in a metric space, A Euclidean bump for a compact set inside an open set).

[F6]

Fréchet–Kolmogorov criterion: a family in $L^{1} (R^{d})$ that is uniformly bounded, has uniformly small tails, and is uniformly translation-continuous is relatively compact, and every sequence in it has a subsequence converging in $L^{1} (R^{d})$ ; the criterion is used with the Axioms of Countable and Dependent Choice, and $L^{1}$ convergence yields an almost-everywhere convergent subsequence, while each $L^{1}$ space is complete (The Fr'echet--Kolmogorov compactness criterion in $L^{p} (R^{n})$ , Convergence in L^1(mu) has an almost-everywhere convergent subsequence, Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

Fubini's theorem selects almost every time slice of an $L^{1} (Π_{T})$ function, compact subsets of $Π_{T}$ are covered by cylinders $\overline{B} (0, l) \times [1 / l, T - 1 / l]$ and exhaustion arguments use Heine–Borel and sequential compactness (Fubini's theorem for L^1 functions on a sigma-finite product, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle, Open ball, closed ball and sphere in a metric space).

Proof

technique · direct

1.1F2F3

Uniform bounds and spatial modulus. By [F2], $∣ u^{j} (t, x) ∣ \leq M$ and $∥ u^{j} (t, \cdot) ∥_{1} \leq ∥ u_{0} ∥_{1}$ for all $j, t$ ; by [F3], $∥ u^{j} (t, \cdot + z) - u^{j} (t, \cdot) ∥_{1} \leq ω_{0} (∣ z ∣)$ with $ω_{0} (r) ↓ 0$ , uniformly in $j$ and $t$ .

1.2F1F2F4F5

The local time modulus. Fix $R > 0$ , $0 < ρ < 1$ , $j$ , and $t, τ$ with $t, t + τ \in [0, T]$ ; put $w = u^{j} (t + τ, \cdot) - u^{j} (t, \cdot)$ . Choose $χ$ as in [F5] and an even mollifier $ϱ_{h}$ with $0 < h < ρ / 4$ ; set $z = ϱ_{h} * w$ and $g = ϱ_{h} * (χ sgn z) \in C_{c}^{\infty} (B_{R + ρ})$ , so that $∥ D^{α} g ∥_{\infty} \leq C h^{- ∣ α ∣}$ for $∣ α ∣ \leq 2$ and $\int_{R^{n}} g w d x = \int_{R^{n}} χ ∣ z ∣ d x$ by [F4]. Assume $τ > 0$ ; negative increments follow by reversing the two times. Since $u^{j}$ solves the viscous equation pointwise at positive times and is $C^{1}$ there, $\frac{d}{d s} \int g u^{j} (t + s, \cdot) d x = \int \nabla g \cdot f_{j} (u^{j} (t + s, \cdot)) d x + ε_{j} \int Δ g u^{j} (t + s, \cdot) d x$ for $s \in [0, τ]$ ; integrating over $s$ and using $∥ f_{j} (v) ∥_{1} \leq L ∥ v ∥_{1}$ for $∣ v ∣ \leq M$ and [F2], $∣ \int g w ∣ \leq \int_{0}^{τ} (∥ \nabla g ∥_{\infty} L ∥ u_{0} ∥_{1} + ε_{j} ∥ Δ g ∥_{\infty} ∥ u_{0} ∥_{1}) d s \leq C_{R} τ h^{- 2}$ .

2.1F3F4step 1.2

The uniform modulus in $L_{l o c}^{1}$ . With $χ, w, z$ as in step 1.2 and [F4], $\int_{B_{R}} ∣ w ∣ \leq \int χ ∣ w ∣ \leq \int χ ∣ z ∣ + \int χ ∣ w - z ∣ \leq C_{R} τ h^{- 2} + 2 ω_{0} (h)$ , because $\int χ ∣ z ∣ = ∣ \int g w ∣$ and $∥ w - z ∥_{L^{1} (B_{R + ρ / 2})} \leq 2 ω_{0} (h)$ by [F3] applied at the two times $t$ and $t + τ$ . Taking $h = τ^{1 / 3}$ (for $0 < τ < (ρ / 4)^{3}$ ) gives $\int_{B_{R}} ∣ u^{j} (t + τ) - u^{j} (t) ∣ \leq C_{R} (ω_{0} (τ^{1 / 3}) + τ^{1 / 3}) = : η_{R} (τ)$ with $η_{R} (τ) \to 0$ as $τ ↓ 0$ , uniformly in $j$ and in $t \in [0, T - τ]$ ; the endpoint $t = 0$ follows by first integrating from a positive time and then using the $L^{1}$ continuity $u^{j} (t) \to u_{0}$ .

3.1F5F6F7step 2.1

Relative compactness on cylinders. Fix $l \geq 1$ and cutoffs $ψ_{l}, θ_{l}$ as in [F5] equal to $1$ on the projections of the cylinder $K_{l} = \overline{B} (0, l) \times [1 / l, T - 1 / l]$ , and set $F_{l}^{j} (t, x) = θ_{l} (t) ψ_{l} (x) u^{j} (t, x)$ , extended by zero. The family $(F_{l}^{j})_{j}$ is uniformly bounded in $L^{1} (R^{n + 1})$ by $T ∥ θ_{l} ψ_{l} ∥_{\infty} ∥ u_{0} ∥_{1}$ , has common compact support (uniform tails), and is uniformly translation-continuous: for a shift $(y, s)$ one has $∥ F_{l}^{j} (\cdot + (y, s)) - F_{l}^{j} ∥_{1} \leq ∥ F_{l}^{j} (t + s, x + y) - F_{l}^{j} (t, x + y) ∥_{1} + ∥ F_{l}^{j} (t, x + y) - F_{l}^{j} (t, x) ∥_{1}$ , where the spatial part is bounded by $T ∥ (θ_{l} ψ_{l}) (\cdot + y) - (θ_{l} ψ_{l}) ∥_{\infty} ∥ u_{0} ∥_{1} + T ∥ θ_{l} ψ_{l} ∥_{\infty} ω_{0} (∣ y ∣)$ and the temporal part by $T ∥ θ_{l} (\cdot + s) - θ_{l} ∥_{\infty} ∥ ψ_{l} ∥_{\infty} ∥ u_{0} ∥_{1} + T ∥ θ_{l} ψ_{l} ∥_{\infty} η_{l^{'}} (∣ s ∣)$ for a radius $l^{'}$ containing $supp ψ_{l}$ , both tending to $0$ as $∣ (y, s) ∣ \to 0$ uniformly in $j$ by step 1.1 and step 2.1. By [F6] every subsequence of $(F_{l}^{j})_{j}$ has a further subsequence converging in $L^{1} (R^{n + 1})$ , hence, after diagonal extraction over the countably many $l \geq 1$ (Dependent Choice), some subsequence of $(u^{j})$ converges in $L^{1} (K_{l})$ for every $l$ , and therefore in $L^{1} (K)$ for every compact $K ⋐ Π_{T}$ , since each such $K$ lies in some $K_{l}$ .

3.2F6F7step 2.1

Continuous representative and the initial trace. Pass to a further subsequence converging almost everywhere on $Π_{T}$ by [F6]. Since $∣ u^{j_{k}} ∣ \leq M$ , this gives $∣ u ∣ \leq M$ almost everywhere; by Fubini and dominated convergence, there is a common full-measure set $S \subset (0, T)$ on which $u^{j_{k}} (t) \to u (t)$ in $L^{1} (B_{m})$ for every integer $m \geq 1$ . Thus $S$ is dense. Fix $m$ and choose an integer $m^{'} > m$ . For $s, t \in S$ sufficiently close that step 2.1 applies, lower semicontinuity on the local ball, followed by its time-modulus estimate on the larger ball, gives $∥ u (t) - u (s) ∥_{L^{1} (B_{m})} \leq {lim inf}_{k} ∥ u^{j_{k}} (t) - u^{j_{k}} (s) ∥_{L^{1} (B_{m})} \leq η_{m^{'}} (∣ t - s ∣) .$ For $t \in S$ sufficiently close to $0$ , the same local comparison with $u^{j_{k}} (0, \cdot) = u_{0}$ gives $∥ u (t) - u_{0} ∥_{L^{1} (B_{m})} \leq η_{m^{'}} (t)$ . This modulus tending to zero at zero makes the map on $S$ uniformly continuous; completeness of $L^{1} (B_{m})$ extends it uniquely to a continuous map on $[0, T]$ , with value $u_{0}$ at $t = 0$ . These extensions agree on nested balls because they agree on the dense set $S$ . Consequently $u$ has a representative in $C ([0, T]; L_{l o c}^{1} (R^{n}))$ with $u (0) = u_{0}$ in $L_{l o c}^{1}$ .

4.1F2F6step 3.1step 3.2

Limit properties. The diagonal limit of step 3.1 lies in $L_{l o c}^{1} (Π_{T})$ and the selected subsequence converges in $L^{1} (K)$ for every compact $K ⋐ Π_{T}$ ; the further almost-everywhere subsequence in step 3.2 preserves these convergences and passes the uniform bound $∣ u^{j_{k}} ∣ \leq M$ to $u$ . This establishes the asserted limit and almost-everywhere convergence.

5.1F2F6step 1.1step 2.1step 3.1∎

Applicability and hypotheses. If $f_{j} \to f$ in $C^{1}$ on compact sets with $\sup_{j} \sup_{∣ s ∣ \leq M + 1} ∣ f_{j}^{'} (s) ∣ < \infty$ — a $C^{1}$ -convergent smooth approximation of one flux — then the hypotheses above hold, so the conclusions apply. The extraction used Dependent Choice in the diagonal step 3.1 (and Countable Choice inside [F6]); the uniform energy dissipation $ε_{j} \int_{0}^{T} ⁣ ⁣ \int ∣ \nabla u^{j} ∣^{2} \leq \frac{1}{2} ∥ u_{0} ∥_{2}^{2}$ supplies only a uniform gradient bound and would not by itself give the compactness in $L^{1}$ obtained from the uniform bounds and translation moduli of steps 1.1 and 2.1.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Global $L^{1}$ contraction from the local estimate

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $n \geq 1$ , $T > 0$ , let $f ⁣ : R \to R^{n}$ be $C^{1}$ and Lipschitz on the common essential range of the solutions below with constant $L \geq 0$ , and let $u, v$ be bounded Kruzhkov entropy solutions on $Π_{T}$ in the sense of Kruzhkov entropy solutions, with initial data $u_{0}, v_{0} \in L^{1} (R^{n}) \cap L^{\infty} (R^{n})$ . Then for almost every $t \in (0, T)$ , $∥ u (t, \cdot) - v (t, \cdot) ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1} .$ If $u, v$ have representatives continuous in $L_{l o c}^{1}$ on $[0, T]$ , the same inequality holds for every $t \in [0, T]$ (Open ball, closed ball and sphere in a metric space, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , $f$ Lipschitz with constant $L \geq 0$ on the common essential range of $u$ and $v$ , bounded Kruzhkov entropy solutions $u, v$ on $Π_{T}$ with $∣ u ∣, ∣ v ∣ \leq M$ almost everywhere, and initial data $u_{0}, v_{0} \in L^{1} (R^{n}) \cap L^{\infty} (R^{n})$ . Set $L_{*} : = \sup_{∣ z ∣ \leq M} ∣ f' (z) ∣ < \infty$ ; in the proof use $L_{*}$ , irrespective of the given constant on the common essential range.

[F1]

Since $f \in C^{1}$ , $L_{*} < \infty$ , and coordinatewise FTC gives $f (b) - f (a) = \int_{a}^{b} f' (z) d z$ and $∣ f (b) - f (a) ∣ \leq L_{*} ∣ b - a ∣$ for $a, b \in [- M, M]$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ). Local $L^{1}$ contraction with this interval constant: for every centre $x_{0}$ and radius $R > 0$ , for almost every $t \in (0, T)$ with $L_{*} t < R$ , $\int_{B (x_{0}, R - L_{*} t)} ∣ u (t, x) - v (t, x) ∣ d x \leq \int_{B (x_{0}, R)} ∣ u_{0} (x) - v_{0} (x) ∣ d x$ , with an exceptional null set depending on $x_{0}, R$ (Local $L^{1}$ contraction for two entropy solutions, Kruzhkov entropy solutions).

[F2]

Monotone convergence for integrals of nonnegative functions over increasing sets: if $0 \leq g_{m} ↑ g$ pointwise then $\int g_{m} ↑ \int g$ ; in particular the integrals of a fixed nonnegative function over the balls $B (0, ρ) ↑ R^{n}$ increase to its integral over $R^{n}$ , finite or infinite (Monotone convergence for the integral).

[F3]

Almost-everywhere assertions concern equivalence classes: a countable union of null sets in $(0, T)$ is null, and members of $L^{1}$ are defined up to modification on null sets (The space $L^{p} (μ)$ as the quotient by null functions, Open ball, closed ball and sphere in a metric space).

Proof

technique · direct

1.1F1

Ball estimates along an exhausting sequence. For $m \geq 1$ put $R_{m} = L_{*} T + m$ , so that $R_{m} > L_{*} t$ for every $t \in (0, T)$ and $R_{m} - L_{*} t = m + L_{*} (T - t) > 0$ . Applying [F1] with centre $0$ and radius $R_{m}$ gives, for every $m$ , an exceptional null set $N_{m} \subseteq (0, T)$ such that for all $t \in (0, T) ∖ N_{m}$ $\int_{B (0, R_{m} - L_{*} t)} ∣ u (t, x) - v (t, x) ∣ d x \leq \int_{B (0, R_{m})} ∣ u_{0} (x) - v_{0} (x) ∣ d x \leq ∥ u_{0} - v_{0} ∥_{1} .$

2.1F2F3step 1.1

Intersection and monotone limit. The set $N = ⋃_{m \geq 1} N_{m}$ is null by [F3]. Fix $t \in (0, T) ∖ N$ , so that the estimates of step 1.1 hold for every $m$ . The balls $B (0, R_{m} - L_{*} t)$ increase to $R^{n}$ as $m ↑ \infty$ , hence the integrals of the fixed nonnegative function $∣ u (t, \cdot) - v (t, \cdot) ∣$ over them increase to $\int_{R^{n}} ∣ u (t, x) - v (t, x) ∣ d x$ , while $\int_{B (0, R_{m})} ∣ u_{0} - v_{0} ∣ ↑ ∥ u_{0} - v_{0} ∥_{1}$ by monotone convergence. Passing to the limit in step 1.1 gives $∥ u (t, \cdot) - v (t, \cdot) ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1} < \infty$ for every $t \in (0, T) ∖ N$ , which is the almost-everywhere assertion and shows that the slice integrals are finite for almost every $t$ .

3.1F2step 2.1∎

The every-time assertion under $L_{l o c}^{1}$ continuity. Assume now that $u$ and $v$ have representatives on $[0, T]$ such that $t_{j} \to t$ implies $u (t_{j}) \to u (t)$ and $v (t_{j}) \to v (t)$ in $L^{1} (K)$ for every compact $K \subseteq R^{n}$ (with one-sided sequences at $t = 0, T$ ). Fix $t \in [0, T]$ and choose $t_{j} \in (0, T) ∖ N$ with $t_{j} \to t$ , possible because $N$ is null. For every fixed ball $B (0, ρ)$ , step 2.1 gives $\int_{B (0, ρ)} ∣ u (t_{j}) - v (t_{j}) ∣ \leq ∥ u (t_{j}) - v (t_{j}) ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ , and $∣ u (t_{j}) - v (t_{j}) ∣ \to ∣ u (t) - v (t) ∣$ in $L^{1} (B (0, ρ))$ because $u (t_{j}) \to u (t)$ and $v (t_{j}) \to v (t)$ there; the inequality $∣ ∥ a ∥_{1} - ∥ b ∥_{1} ∣ \leq ∥ a - b ∥_{1}$ gives convergence of these integrals directly. Hence $\int_{B (0, ρ)} ∣ u (t) - v (t) ∣ \leq {lim inf}_{j} \int_{B (0, ρ)} ∣ u (t_{j}) - v (t_{j}) ∣ \leq ∥ u_{0} - v_{0} ∥_{1}$ . Letting $ρ ↑ \infty$ and using monotone convergence once more gives $∥ u (t) - v (t) ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Existence of bounded Kruzhkov entropy solutions

Statement

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the heat-kernel, $L^{1}$ completeness and vanishing-viscosity extraction interfaces used below. Let $n \geq 1$ , $T > 0$ , let $f ⁣ : R \to R^{n}$ be locally Lipschitz and $C^{1}$ , and let $u_{0} \in L^{\infty} (R^{n}) \cap L^{1} (R^{n})$ . Then there exists a Kruzhkov entropy solution $u \in L^{\infty} (Π_{T}) \cap C^{0} ([0, T]; L_{l o c}^{1} (R^{n}))$ of $u_{t} + {div}_{x} f (u) = 0$ with $u (\cdot, 0) = u_{0}$ in the strong $L_{l o c}^{1}$ sense, satisfying $∥ u (t, \cdot) ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ for every $t$ . By Uniqueness, comparison and order preservation of entropy solutions this solution is unique. The route is vanishing viscosity, and the compactness comes from the translation lemmas, not from the energy dissipation alone (Scalar conservation laws, fluxes and Cauchy data, Kruzhkov entropy solutions).

Facts & Assumptions

Given: Countable and Dependent Choice, $n \geq 1$ , $T > 0$ , a locally Lipschitz $C^{1}$ flux $f$ , and $u_{0} \in L^{\infty} (R^{n}) \cap L^{1} (R^{n})$ with $M = ∥ u_{0} ∥_{\infty}$ .

[F1]

The weak formulation: $u$ is a distributional weak solution of $u_{t} + {div}_{x} f (u) = 0$ on $Π_{T}$ iff $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla φ) = 0$ for every $φ \in C_{c}^{\infty} (Π_{T})$ ; subtracting the constant $f (0)$ from the flux changes neither the divergence term nor the Kruzhkov fluxes $sgn (u - k) (f (u) - f (k))$ , so all existence and entropy statements may be proved for the normalized flux $\tilde{f} = f - f (0)$ and transferred back (Scalar conservation laws, fluxes and Cauchy data, Kruzhkov entropy solutions).

[F2]

Viscous solutions: for every $C^{2}$ flux $g$ with $g (0) = 0$ , every $ε \in (0, 1]$ and every datum in $C_{c}^{\infty}$ , there is a global classical solution of $u_{t} + {div}_{x} g (u) = ε Δ u$ with $u \in C ([0, T]; C_{b}) \cap C ([0, T]; L^{1}) \cap C^{1, 2} (R^{n} \times (0, T))$ , range contained in the initial range, and $∥ u (t, \cdot) ∥_{1} \leq ∥ u_{0} ∥_{1}$ (The viscous scalar Cauchy problem with smooth data has a global classical solution, Uniform L-infinity, mass and energy bounds for the viscous approximations).

[F3]

Viscous entropy balance: for every convex $C^{2}$ entropy $η$ with flux $q^{'} = η^{'} g^{'}$ , $\partial_{t} η (u) + {div}_{x} q (u) = ε Δ η (u) - ε η^{''} (u) ∣ \nabla u ∣^{2}$ pointwise, and the Laplacian term integrates by parts against compactly supported tests (The viscous entropy dissipation identity).

[F4]

Mollification: convolving a locally integrable function with a radial mollifier gives a smooth function; the mollified derivatives are the convolutions of the derivatives, and on compact sets the mollified flux and its derivative converge uniformly to the original for $C^{1}$ data; approximate identities converge in $L^{1}$ , and the classes of $L^{p}$ are equivalence classes (A unit-mass smooth bump generates an $L^{1}$ approximate identity, Convolution with a mollifier is smooth, and derivatives pass under the integral sign, The space $L^{p} (μ)$ as the quotient by null functions). The norm convergence assertion is Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ; the compact cutoffs are A Euclidean bump for a compact set inside an open set.

[F5]

Vanishing-viscosity compactness: for smooth compactly supported data, $C^{2}$ fluxes $f_{m}$ with $f_{m} (0) = 0$ and uniformly bounded derivatives on the range $[- M - 1, M + 1]$ , and $0 < ε_{m} \leq 1$ , $ε_{m} ↓ 0$ , the viscous solutions have a subsequence converging in $L^{1} (K)$ for every compact $K ⋐ Π_{T}$ and almost everywhere on $Π_{T}$ to $u \in L^{\infty} \cap L_{l o c}^{1} (Π_{T})$ with $∥ u ∥_{\infty} \leq M$ , having a representative in $C ([0, T]; L_{l o c}^{1})$ with $u (0) = u_{0}$ in $L_{l o c}^{1}$ ; the extraction uses Dependent Choice (Vanishing-viscosity families are locally precompact in $L^{1}$ ).

[F6]

Contraction and uniqueness: for data in $L^{1} \cap L^{\infty}$ the global $L^{1}$ difference of two entropy solutions is bounded by the $L^{1}$ difference of their data, at every time if the representatives are $L_{l o c}^{1}$ -continuous; two bounded Kruzhkov entropy solutions with the same data coincide almost everywhere (Global $L^{1}$ contraction from the local estimate, Uniqueness, comparison and order preservation of entropy solutions).

[F7]

Limits and completeness: dominated and monotone convergence for integrals over fixed compact sets with uniformly bounded integrands, and completeness of $L^{1} (K)$ for compact $K$ (Dominated convergence, Monotone convergence for the integral, Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ ).

Proof

technique · direct

1.1F1F4

Normalization and smooth flux approximation. By [F1] it suffices to treat $\tilde{f} = f - f (0)$ . Choose a radial mollifier and, for $m \geq 1$ , set $f_{m} (s) = χ (s) (\tilde{f} * ρ_{1 / m}) (s) - (\tilde{f} * ρ_{1 / m}) (0)$ , where $χ \in C_{c}^{\infty}$ equals $1$ on $[- M - \frac{1}{2}, M + \frac{1}{2}]$ and is supported in $[- M - 1, M + 1]$ . Then each $f_{m}$ is $C^{\infty}$ (hence $C^{2}$ ), $f_{m} (0) = 0$ , and on $[- M - \frac{1}{2}, M + \frac{1}{2}]$ one has $f_{m} \to \tilde{f}$ and $f_{m}^{'} = {\tilde{f}}^{'} * ρ_{1 / m} \to {\tilde{f}}^{'}$ uniformly by [F4]; in particular $\sup_{m} \sup_{∣ s ∣ \leq M + 1} ∣ f_{m}^{'} (s) ∣ < \infty$ .

1.2F2F5

Smooth-datum case: extraction. Let $u_{0} \in C_{c}^{\infty} (R^{n})$ with $∥ u_{0} ∥_{\infty} = M$ , choose $ε_{m} ↓ 0$ with $0 < ε_{m} \leq 1$ , and let $u^{m}$ be the global classical solution with flux $f_{m}$ and datum $u_{0}$ given by [F2]. Then $∣ u^{m} (t, x) ∣ \leq ∥ u_{0} ∥_{\infty} = M$ by the range bound of [F2], so the hypotheses of [F5] are met; passing to a subsequence (relabelled) there is $u \in L^{\infty} \cap L_{l o c}^{1} (Π_{T})$ with $∥ u ∥_{\infty} \leq M$ , $u^{m} \to u$ in $L^{1} (K)$ for every compact $K ⋐ Π_{T}$ and almost everywhere, and $u$ has a representative in $C ([0, T]; L_{l o c}^{1})$ with $u (0) = u_{0}$ in $L_{l o c}^{1}$ . At every time $t$ this representative obeys $∣ u (t, \cdot) ∣ \leq M$ almost everywhere: for $t$ in the full-measure set where the construction of [F5] passes the bound, slicewise almost-everywhere convergence preserves it, and general $t$ follows by $L_{l o c}^{1}$ -continuity.

2.1F1F7step 1.1step 1.2

The weak equation passes to the limit. For $φ \in C_{c}^{\infty} (Π_{T})$ , testing the pointwise viscous equation of $u^{m}$ gives $\int_{Π_{T}} (u^{m} φ_{t} + f_{m} (u^{m}) \cdot \nabla φ) = - ε_{m} \int_{Π_{T}} u^{m} Δ φ$ , whose right side is bounded by $ε_{m} M ∥ Δ φ ∥_{1} \to 0$ . The left side converges to $\int_{Π_{T}} (u φ_{t} + \tilde{f} (u) \cdot \nabla φ)$ by [F7], because $u^{m} \to u$ almost everywhere with uniform bounds and $f_{m} \to \tilde{f}$ uniformly on $[- M, M]$ ; hence $u$ is a weak solution for $\tilde{f}$ , and therefore for $f$ by [F1].

2.2F3F4F7step 1.1step 1.2

Entropy inequalities for $∣ k ∣ \leq M$ . Fix $k \in [- M, M]$ and $δ > 0$ , and put $η_{δ} (r) = \sqrt{r^{2} + δ^{2}} - δ$ , a convex $C^{2}$ function with $0 \leq η_{δ} \leq ∣ \cdot ∣$ , $∣ η_{δ} (r) - ∣ r ∣ ∣ \leq δ$ , and $η_{δ}^{''} \geq 0$ ; let $q_{δ, k, m} (s) = \int_{k}^{s} η_{δ}^{'} (r - k) f_{m}^{'} (r) d r$ , so $q_{δ, k, m}^{'} = η_{δ}^{'} (\cdot - k) f_{m}^{'}$ . Let $φ \in C_{c}^{\infty} (R^{n} \times [0, T))$ be nonnegative. Multiplying the exact viscous balance [F3] for the pair $(η_{δ} (\cdot - k), q_{δ, k, m})$ by $φ$ , integrating over $Π_{T}$ and integrating the Laplacian by parts gives $I_{m, δ} : = \int_{Π_{T}} (η_{δ} (u^{m} - k) φ_{t} + q_{δ, k, m} (u^{m}) \cdot \nabla φ) + \int_{R^{n}} η_{δ} (u_{0} - k) φ (x, 0) d x = - ε_{m} \int_{Π_{T}} η_{δ} (u^{m} - k) Δ φ + ε_{m} \int_{Π_{T}} η_{δ}^{''} (u^{m} - k) ∣ \nabla u^{m} ∣^{2} φ$ — the only boundary term is the one at $t = 0$ , displayed with the initial datum; the last term is nonnegative and the first is bounded by $ε_{m} \cdot 2 M ∥ Δ φ ∥_{1} \to 0$ , so ${lim inf}_{m} I_{m, δ} \geq 0$ . On the other hand $I_{m, δ}$ converges by [F7]: $u^{m} \to u$ almost everywhere, $∣ η_{δ} (u^{m} - k) ∣ \leq 2 M$ , and $q_{δ, k, m} \to q_{δ, k}$ uniformly on $[- M, M]$ , where $q_{δ, k} (s) = \int_{k}^{s} η_{δ}^{'} (r - k) {\tilde{f}}^{'} (r) d r$ , so the limit obeys $I_{δ} = \int_{Π_{T}} (η_{δ} (u - k) φ_{t} + q_{δ, k} (u) \cdot \nabla φ) + \int_{R^{n}} η_{δ} (u_{0} - k) φ (x, 0) d x \geq 0$ .

3.1F7step 2.1step 2.2

The Kruzhkov inequalities. Letting $δ ↓ 0$ in step 2.2, $η_{δ} (r - k) \to ∣ r - k ∣$ uniformly on $[- M, M]$ and $q_{δ, k} (s) \to \int_{k}^{s} sgn (r - k) {\tilde{f}}^{'} (r) d r = sgn (s - k) (\tilde{f} (s) - \tilde{f} (k)) = q_{k} (s)$ uniformly on $[- M, M]$ by dominated convergence, since $∣ η_{δ}^{'} ∣ \leq 1$ and ${\tilde{f}}^{'}$ is continuous there; hence dominated convergence gives $\int_{Π_{T}} (∣ u - k ∣ φ_{t} + q_{k} (u) \cdot \nabla φ) + \int ∣ u_{0} - k ∣ φ (x, 0) d x \geq 0$ for every nonnegative $φ \in C_{c}^{\infty} (R^{n} \times [0, T))$ and every $k \in [- M, M]$ . If $k > M$ , then $u - k < 0$ almost everywhere, $η_{k}$ is affine on the range $[- M, M]$ , and $q_{k} (s) = \tilde{f} (k) - \tilde{f} (s)$ , so the identity $\partial_{t} η_{k} (u) + {div}_{x} q_{k} (u) = - (u_{t} + {div}_{x} \tilde{f} (u)) = 0$ holds by step 2.1, and similarly for $k < - M$ ; thus all Kruzhkov inequalities hold and, with the trace of step 1.2, $u$ is a bounded Kruzhkov entropy solution with datum $u_{0}$ .

4.1F4F6step 2.1step 2.2step 3.1

General datum: approximation and Cauchy property. Now let $u_{0} \in L^{\infty} \cap L^{1}$ be arbitrary. By [F4] choose $u_{0}^{r} \in C_{c}^{\infty}$ with $∥ u_{0}^{r} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ and $u_{0}^{r} \to u_{0}$ in $L^{1}$ (truncate $u_{0}$ to a large ball and mollify). Steps 1.1–3.1 applied to each smooth datum $u_{0}^{r}$ give bounded Kruzhkov entropy solutions $u^{r}$ for the flux $f$ , with $L_{l o c}^{1}$ -continuous representatives and $∥ u^{r} (t, \cdot) ∥_{\infty} \leq ∥ u_{0}^{r} ∥_{\infty} \leq M$ . By [F6], for every $t \in [0, T]$ , $∥ u^{r} (t) - u^{ℓ} (t) ∥_{1} \leq ∥ u_{0}^{r} - u_{0}^{ℓ} ∥_{1}$ , so $\sup_{t \in [0, T]} ∥ u^{r} (t) - u^{ℓ} (t) ∥_{L^{1} (K)} \leq ∥ u_{0}^{r} - u_{0}^{ℓ} ∥_{1}$ for every compact $K$ , and the right side tends to $0$ as $r, ℓ \to \infty$ .

5.1F7step 3.1step 4.1

The limit for general datum. By completeness of $L^{1} (K)$ [F7] and the uniform-in-time contraction in step 4.1, $u^{r}$ converges in $C ([0, T]; L^{1} (K))$ for each compact ball $K$ , consistently on nested balls, to $u \in C ([0, T]; L_{l o c}^{1} (R^{n}))$ . Since $u^{r} (0) = u_{0}^{r} \to u_{0}$ in $L^{1}$ , this representative has initial trace $u_{0}$ . For every $t$ and ball $K$ , the inequality $(∣ u (t) ∣ - M)_{+} \leq ∣ u (t) - u^{r} (t) ∣ + (∣ u^{r} (t) ∣ - M)_{+}$ and $∥ u^{r} (t) ∥_{\infty} \leq M$ show, after integration on $K$ and passage to the $L^{1} (K)$ limit, that $(∣ u (t) ∣ - M)_{+} = 0$ almost everywhere there. Thus $∥ u (t) ∥_{\infty} \leq M$ for every $t$ . The weak equation passes to the limit because $f$ is Lipschitz on $[- M, M]$ and $u^{r} \to u$ in local $L^{1}$ . For each fixed $k \in R$ , $s \mapsto ∣ s - k ∣$ is $1$ -Lipschitz and $q_{k} (s) = sgn (s - k) (f (s) - f (k))$ is Lipschitz on $[- M, M]$ with constant at most $\sup_{∣ s ∣ \leq M} ∣ f^{'} (s) ∣$ . Therefore the entropy and flux terms in the inequality of step 3.1 converge in $L^{1}$ on every test support; the initial entropy term converges by $u_{0}^{r} \to u_{0}$ in $L^{1}$ and the same Lipschitz bound for $∣ \cdot - k ∣$ . Passing to the limit proves every Kruzhkov inequality without requiring an almost-everywhere subsequence for the general-data approximation. Hence $u$ is a bounded Kruzhkov entropy solution with datum $u_{0}$ .

6.1F6step 1.2step 5.1∎

Uniqueness and conclusion. If $v$ is another bounded Kruzhkov entropy solution with the same datum $u_{0}$ , then $u = v$ almost everywhere on $Π_{T}$ by [F6], so the constructed solution is the unique bounded Kruzhkov entropy solution with this datum; this completes the proof.

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The convex entropy condition for a single shock is the chord condition

Statement

Let $n = 1$ , $f \in C^{1} (R)$ , and let $u$ be a piecewise $C^{1}$ weak solution with a single jump from the left state $u^{-}$ to the right state $u^{+}$ across a $C^{1}$ curve $x = s (t)$ whose speed satisfies the Rankine--Hugoniot condition $s^{'} = (f (u^{+}) - f (u^{-})) / (u^{+} - u^{-})$ . Put $F (z) = f (z) - f (u^{-}) - s^{'} (z - u^{-}),$ so that $F (u^{-}) = F (u^{+}) = 0$ . Then the entropy inequality $η (u)_{t} + q (u)_{x} \leq 0$ of Convex entropy--entropy flux pairs holds for every convex entropy pair $(η, q)$ if and only if $F (z) (u^{+} - u^{-}) \geq 0 for every z between u^{-} and u^{+};$ equivalently, in the case $u^{-} < u^{+}$ the graph of $f$ on $[u^{-}, u^{+}]$ lies above the chord joining $(u^{-}, f (u^{-}))$ and $(u^{+}, f (u^{+}))$ , while in the case $u^{-} > u^{+}$ it lies below that chord, both in the non-strict sense (Piecewise smooth shocks and one-sided traces, Convex and strictly convex functions on Euclidean convex sets).

Facts & Assumptions

Given: $n = 1$ , $f \in C^{1}$ , a single-jump piecewise $C^{1}$ weak solution with states $u^{-} \neq u^{+}$ and speed $s^{'}$ satisfying Rankine--Hugoniot, and an arbitrary convex entropy pair $(η, q)$ with $η \in C^{2}$ and $q^{'} = η^{'} f^{'}$ .

[F1]

The jump configuration and Rankine--Hugoniot condition are as in Piecewise smooth shocks and one-sided traces and The Rankine--Hugoniot jump condition in space--time normal form: in one dimension the interface is a graph $x = s (t)$ with minus side $x < s (t)$ , plus side $x > s (t)$ , unit normal $ν = (- s^{'}, 1) / \sqrt{1 + s^{' 2}}$ , and $s^{'} (u^{+} - u^{-}) = f (u^{+}) - f (u^{-})$ .

[F2]

The graph integration of The Rankine--Hugoniot jump condition in space--time normal form, applied to $(η (u), q (u))$ , gives the interface production $([q] - s^{'} [η]) δ (x - s (t))$ , where the latter distribution pairs by $\int φ (t, s (t)) d t$ . For smooth pairs the production vanishes in the classical side regions by the chain rule. Smooth nonnegative bumps can be placed on any interface patch (Explicit compactly supported smooth cutoffs). Thus the entropy inequality is equivalent to nonpositive jump production at every point (Convex entropy--entropy flux pairs, Kruzhkov entropy solutions).

[F3]

Primitives: since $η^{'}$ and $f^{'}$ are continuous, $q (z) = \int_{0}^{z} η^{'} (r) f^{'} (r) d r$ up to a constant, and increments of $C^{1}$ functions are integrals of their derivatives; the fundamental theorem of calculus, its use under limits, and the primitive construction are as in Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ and The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ .

[F4]

Approximation tools: monotone bounded convergence for limits of test functions and dominated convergence for the passing of inequalities (Monotone convergence for the integral, Dominated convergence, Absolute value in an ordered field).

Proof

technique · direct

1.1F2F3

Jump entropy production. By [F2] the entropy inequality for $(η, q)$ is equivalent to $[q] - s^{'} [η] \leq 0$ . Using [F3] in the orientation of the jump, $[q] = \int_{u^{-}}^{u^{+}} q^{'} (z) d z = \int_{u^{-}}^{u^{+}} η^{'} (z) f^{'} (z) d z$ and $[η] = \int_{u^{-}}^{u^{+}} η^{'} (z) d z$ , so the condition is $\int_{u^{-}}^{u^{+}} η^{'} (z) (f^{'} (z) - s^{'}) d z \leq 0$ .

2.1F1F3step 1.1

Smooth-pair sufficiency. At a fixed interface point put $a = u^{-}$ , $b = u^{+}$ and $s^{'} = [f] / [u]$ . Integration by parts, with $F (a) = F (b) = 0$ , gives $[q] - s^{'} [η] = \int_{a}^{b} η^{'} (z) F^{'} (z) d z = - \int_{a}^{b} η^{''} (z) F (z) d z .$ If $a < b$ and $F \geq 0$ , this is nonpositive since $η^{''} \geq 0$ . If $a > b$ and $F \leq 0$ , reversal of the integral gives the same conclusion.

3.1F1F2F3step 2.1

Necessity. If $a < b$ and $F (z_{0}) < 0$ for some interior $z_{0}$ , continuity supplies an interval on which $F < 0$ . Choose a smooth nonnegative bump $β$ supported there and not identically zero, and define $η^{'} (z) = \int_{0}^{z} β (r) d r$ , $η (z) = \int_{0}^{z} η^{'} (r) d r$ . Then $η^{''} = β \geq 0$ , so this is a smooth convex entropy; step 2.1 gives strictly positive production, a contradiction. If $a > b$ and $F (z_{0}) > 0$ , the same bump and reversed integral again give positive production. Thus all smooth convex inequalities force $F (z) (b - a) \geq 0$ . For a Kruzhkov pair with $k$ between the states, a direct subtraction gives $[q_{k}] - s^{'} [η_{k}] = - 2 sgn (b - a) F (k)$ ; outside the interval it is zero. This also proves exact equivalence with the Kruzhkov jump criterion.

4.1F3F4step 2.1step 3.1∎

Nonsmooth pairs and chord interpretation. A finite convex entropy is uniformly approximated on compact intervals by its convolution with a nonnegative smooth unit-mass bump at scale $δ$ . These convolutions are smooth and convex (average the convexity inequality), and their derivatives converge at each differentiability point of $η$ , while remaining bounded by a common local Lipschitz constant. The integral fluxes therefore converge uniformly by dominated convergence, so the smooth entropy inequalities of step 2.1 pass to every locally Lipschitz convex pair, both in the side regions and at the jump. Together with step 3.1 this proves the equivalence. Finally $F (z) \geq 0$ means $f (z)$ lies above $f (a) + s^{'} (z - a)$ for $a < b$ ; $F (z) \leq 0$ means it lies below for $a > b$ . This line is the chord through the two states.

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The self-similar Riemann problem

Definition

Let $n = 1$ and $f \in C^{1} (R)$ (Scalar conservation laws, fluxes and Cauchy data). The Riemann problem for $u_{t} + f (u)_{x} = 0$ prescribes the two-state initial datum $u_{0} (x) = {\begin{cases} u_{L}, & x < 0, \\ u_{R}, & x > 0, \end{cases} u_{L}, u_{R} \in R .$ One looks for self-similar solutions $u (t, x) = v (x / t)$ , $t > 0$ , that is, solutions invariant under the scaling $(t, x) \mapsto (λ t, λ x)$ , $λ > 0$ ; such a $u$ is determined by the single function $v ⁣ : R \to R$ and is constant on each ray $x = ξ t$ .

The initial condition is read as the strong local $L^{1}$ trace $u (t, \cdot) \to u_{0}$ as $t ↓ 0$ (Kruzhkov entropy solutions), and any jump or corner of $v$ occurs on a ray; admissibility is the entropy condition of Kruzhkov entropy solutions, not a further restriction on the self-similar ansatz. Self-similarity is an ansatz to be justified by the uniqueness theorem rather than an additional hypothesis. No choice principle occurs.

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The Riemann solver for a strictly convex flux

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $f \in C^{2} (R)$ be strictly convex and let $u_{L}, u_{R} \in R$ . The unique Kruzhkov entropy solution of the Riemann problem (The self-similar Riemann problem) is:

(i) if $u_{L} > u_{R}$ , the shock $u (t, x) = {\begin{cases} u_{L}, & x < s t, \\ u_{R}, & x > s t, \end{cases} s = \frac{f (u_{R}) - f (u_{L})}{u_{R} - u_{L}};$

(ii) if $u_{L} < u_{R}$ , the centred rarefaction $u (t, x) = {\begin{cases} u_{L}, & x / t \leq f^{'} (u_{L}), \\ (f^{'})^{- 1} (x / t), & f^{'} (u_{L}) < x / t < f^{'} (u_{R}), \\ u_{R}, & x / t \geq f^{'} (u_{R}) . \end{cases}$

Here $f^{'}$ is continuous and strictly increasing, so its inverse on $[f^{'} (u_{L}), f^{'} (u_{R})]$ is continuous; it need not be differentiable, and the rarefaction may have a cusp when $f^{''}$ vanishes. Both profiles satisfy the weak conservation law, all Kruzhkov entropy inequalities, and the strong local $L^{1}$ initial trace. Uniqueness in the bounded Kruzhkov class follows from Uniqueness, comparison and order preservation of entropy solutions. If $u_{L} = u_{R}$ , the constant solution is the unique one (Scalar conservation laws, fluxes and Cauchy data, Kruzhkov entropy solutions).

Facts & Assumptions

Given: Countable Choice, a strictly convex flux $f \in C^{2} (R)$ , states $u_{L}, u_{R} \in R$ , the Riemann datum $u_{0}$ , and the self-similar profiles of the statement.

[F1]

The interior weak equation and the self-similar ansatz: the interior distributional equation is equivalent to $\int_{Π_{T}} (u φ_{t} + f (u) φ_{x}) = 0$ for every $φ \in C_{c}^{\infty} (Π_{T})$ , and a self-similar solution has the form $u (t, x) = U (x / t)$ , constant along rays (Scalar conservation laws, fluxes and Cauchy data, The self-similar Riemann problem).

[F2]

Interface computation at a single jump: for a piecewise $C^{1}$ function with one interface and speed $s$ , the weak residual against a test function supported near the interface equals $- \int_{Γ} φ ([u] ν_{t} + [f (u)] ν_{x}) d S$ ; in one dimension with the graph $x = s t$ , $[u] ν_{t} + [f] ν_{x} = ([f] - s [u]) / \sqrt{1 + s^{2}}$ . Thus Rankine--Hugoniot $s [u] = [f]$ makes the residual vanish there, and across a continuous interface (equal traces of $u$ , hence of $f (u)$ ) the contribution vanishes identically (The Rankine--Hugoniot jump condition in space--time normal form).

[F3]

Chord criterion at a jump: a piecewise $C^{1}$ weak solution with a single nontrivial jump of speed $s$ satisfies the entropy inequality for every convex $C^{2}$ pair if and only if $F (z) (u^{+} - u^{-}) \geq 0$ for all $z$ between the states, where $F (z) = f (z) - f (u^{-}) - s (z - u^{-})$ (The convex entropy condition for a single shock is the chord condition).

[F4]

Strict convexity: $f^{'}$ is strictly increasing by the secant argument in The Lax shock inequalities for convex scalar laws, so $f^{'} (u_{L}) < f^{'} (u_{R})$ when $u_{L} < u_{R}$ and the inverse $(f^{'})^{- 1} ⁣ : [f^{'} (u_{L}), f^{'} (u_{R})] \to [u_{L}, u_{R}]$ is continuous, strictly increasing; the graph of $f$ lies strictly below every chord on the interior of its interval (Convex and strictly convex functions on Euclidean convex sets, A differentiable function on an open interval is convex if and only if its derivative is nondecreasing).

[F5]

Calculus and regularization: the chain rule and algebra of derivatives compute the classical residual of a $C^{1}$ self-similar profile; for $f_{ϵ} (r) = f (r) + \frac{ϵ}{2} r^{2}$ one has $f_{ϵ}^{''} \geq ϵ$ , so $f_{ϵ}^{'}$ is a $C^{1}$ diffeomorphism of $[u_{L}, u_{R}]$ onto its image and $ψ_{ϵ} = (f_{ϵ}^{'})^{- 1}$ is $C^{1}$ on $[f_{ϵ}^{'} (u_{L}), f_{ϵ}^{'} (u_{R})]$ by the inverse function theorem; on the fixed interval $[u_{L}, u_{R}]$ the derivatives $f_{ϵ}^{'}$ converge uniformly to $f^{'}$ as $ϵ ↓ 0$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Euclidean inverse function theorem).

[F6]

Limits: dominated convergence and uniform convergence on compact sets justify passing to the limit in the weak and entropy test integrals, and $L^{p}$ membership is a property of equivalence classes (Dominated convergence, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F1F2given

The shock profile solves the equation and has the right trace. Assume $u_{L} > u_{R}$ and let $u$ be the profile (i) with speed $s = (f (u_{R}) - f (u_{L})) / (u_{R} - u_{L})$ , so $s [u] = [f]$ with $[h] = h (u_{R}) - h (u_{L})$ : this is the Rankine--Hugoniot condition. By [F2] the weak residual of a piecewise constant profile with a single jump reduces to the interface integral $- \int φ (t, s t) ([f] - s [u]) d t$ , which vanishes; hence $u$ is a distributional weak solution. Moreover $u (t, \cdot)$ differs from $u_{0}$ only on the interval between $0$ and $s t$ , whose length is $∣ s ∣ t$ , and $∣ u - u_{0} ∣ \leq ∣ u_{L} - u_{R} ∣$ there, so $\int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x \leq ∣ u_{L} - u_{R} ∣ ∣ s ∣ t \to 0$ as $t ↓ 0$ : the strong local $L^{1}$ trace holds.

1.2F2F5

Regularized fans are weak solutions with vanishing entropy production. For $ϵ > 0$ put $f_{ϵ} (r) = f (r) + \frac{ϵ}{2} r^{2}$ , let $ψ_{ϵ} = (f_{ϵ}^{'})^{- 1}$ on $[f_{ϵ}^{'} (u_{L}), f_{ϵ}^{'} (u_{R})]$ , and define $U_{ϵ}$ by the same three-branch formula with $ξ_{L}^{ϵ} = f_{ϵ}^{'} (u_{L})$ , $ξ_{R}^{ϵ} = f_{ϵ}^{'} (u_{R})$ , and $u^{ϵ} (t, x) = U_{ϵ} (x / t)$ . Each branch is $C^{1}$ by [F5], and on the open middle region the chain rule gives $u_{t}^{ϵ} + f_{ϵ} (u^{ϵ})_{x} = \frac{1}{t} ψ_{ϵ}^{'} (ξ) (f_{ϵ}^{'} (ψ_{ϵ} (ξ)) - ξ) = 0$ since $f_{ϵ}^{'} (ψ_{ϵ} (ξ)) = ξ$ ; on the outer regions the profile is constant, so the residual vanishes pointwise there as well. At the two interfaces the traces of $u^{ϵ}$ , hence of $f_{ϵ} (u^{ϵ})$ , agree from both sides, so by [F2] no interface term arises and $u^{ϵ}$ is a distributional weak solution of $u_{t} + f_{ϵ} (u)_{x} = 0$ . The same computation applied to a convex $C^{2}$ pair $(η, q_{ϵ})$ with $q_{ϵ}^{'} = η^{'} f_{ϵ}^{'}$ gives $\partial_{t} η (u^{ϵ}) + \partial_{x} q_{ϵ} (u^{ϵ}) = η^{'} (u^{ϵ}) (u_{t}^{ϵ} + f_{ϵ}^{'} (u^{ϵ}) u_{x}^{ϵ}) = 0$ on each open branch and continuous traces $η (u^{ϵ}), q_{ϵ} (u^{ϵ})$ at the interfaces, so the entropy residual is identically $0$ for every convex $C^{2}$ pair.

2.1F3F4step 1.1

The shock is entropic. With $u^{-} = u_{L}$ , $u^{+} = u_{R}$ and speed $s$ , the function $F (z) = f (z) - f (u_{L}) - s (z - u_{L})$ satisfies $F (u_{L}) = F (u_{R}) = 0$ , and by strict convexity [F4] the graph of $f$ lies strictly below the chord through $(u_{L}, f (u_{L}))$ , $(u_{R}, f (u_{R}))$ on $(u_{R}, u_{L})$ ; that chord has slope $s$ , so $F (z) < 0$ for $z \in (u_{R}, u_{L})$ . Since $u_{R} - u_{L} < 0$ , $F (z) (u_{R} - u_{L}) \geq 0$ on $[u_{R}, u_{L}]$ , and the chord criterion [F3] gives the entropy inequality for every convex $C^{2}$ pair.

2.2F1F4F5

The rarefaction profile and its inverse. Assume $u_{L} < u_{R}$ and set $ξ_{L} = f^{'} (u_{L}) < ξ_{R} = f^{'} (u_{R})$ ; by [F4] the inverse $ψ = (f^{'})^{- 1} ⁣ : [ξ_{L}, ξ_{R}] \to [u_{L}, u_{R}]$ is continuous and strictly increasing. Define $U (ξ) = u_{L}$ for $ξ \leq ξ_{L}$ , $U (ξ) = ψ (ξ)$ for $ξ_{L} \leq ξ \leq ξ_{R}$ , and $U (ξ) = u_{R}$ for $ξ \geq ξ_{R}$ , and set $u (t, x) = U (x / t)$ for $t > 0$ . Then $u$ takes values in $[u_{L}, u_{R}]$ , differs from $u_{0}$ only between $\min {0, ξ_{L}} t$ and $\max {0, ξ_{R}} t$ , an interval of length $(\max {0, ξ_{R}} - \min {0, ξ_{L}}) t$ , and satisfies the strong local $L^{1}$ initial trace by the same estimate as in step 1.1.

2.3F6step 1.2

Passage to the limiting fan. As $ϵ ↓ 0$ , $f_{ϵ}^{'} \to f^{'}$ uniformly on $[u_{L}, u_{R}]$ by [F5], so the clamped inverse profiles converge uniformly on $R$ . Indeed, if $D = \max {∣ u_{L} ∣, ∣ u_{R} ∣}$ , then $∣ f_{ϵ}^{'} - f^{'} ∣ \leq ϵ D$ on the state interval, so $U (ξ - ϵ D) \leq U_{ϵ} (ξ) \leq U (ξ + ϵ D)$ . The extended continuous profile $U$ is uniformly continuous (it is constant outside a compact interval), yielding $\sup_{ξ} ∣ U_{ϵ} - U ∣ \to 0$ ; also $f_{ϵ} \to f$ and $q_{ϵ} \to q$ uniformly on the compact range $[u_{L}, u_{R}]$ , where $q^{'} = η^{'} f^{'}$ is the flux of the same convex pair for $f$ . Passing to the limit in the weak residual: $∣ \int_{Π_{T}} (U_{ϵ} (x / t) - U (x / t)) φ_{t} ∣ \leq \sup ∣ U_{ϵ} - U ∣ ∥ φ_{t} ∥_{1} \to 0$ and $∥ f_{ϵ} (U_{ϵ}) - f (U) ∥_{\infty} \leq ∥ f_{ϵ} - f ∥_{\infty, [u_{L}, u_{R}]} + L i p (f) \sup ∣ U_{ϵ} - U ∣ \to 0$ , so $\int_{Π_{T}} (U φ_{t} + f (U) φ_{x}) = 0$ and $u$ is a weak solution. The entropy residual passes similarly, so for every convex $C^{2}$ pair $(η, q)$ with $q^{'} = η^{'} f^{'}$ one has $\int_{Π_{T}} (η (u) φ_{t} + q (u) φ_{x}) \geq 0$ for every nonnegative $φ \in C_{c}^{\infty} (Π_{T})$ .

3.1F6step 2.1step 2.3

Kruzhkov pairs by smoothing. Fix $k \in R$ and let $η^{δ} (r) = \sqrt{(r - k)^{2} + δ^{2}} - δ$ , a smooth convex function with $0 \leq η^{δ} (r) \leq ∣ r - k ∣$ and $∣ η^{δ} (r) - ∣ r - k ∣ ∣ \leq δ$ , and let $q^{δ} (s) = \int_{k}^{s} (η^{δ})^{'} (r) f^{'} (r) d r$ , so that $(q^{δ})^{'} = (η^{δ})^{'} f^{'}$ . Since $z \mapsto sgn (z - k)$ is bounded and $f^{'}$ is continuous on the compact range of the profiles, dominated convergence gives $q^{δ} (s) \to \int_{k}^{s} sgn (r - k) f^{'} (r) d r = sgn (s - k) (f (s) - f (k)) = q_{k} (s)$ uniformly for $s$ in the profile range; and $η^{δ} \to η_{k}$ uniformly there. Applying the entropy inequality of step 2.1 (shock case, via [F3]) or of step 2.3 (rarefaction case) to $(η^{δ}, q^{δ})$ and passing to the limit using uniform convergence and $φ \in C_{c}^{\infty}$ gives $\int_{Π_{T}} (η_{k} (u) φ_{t} + q_{k} (u) φ_{x}) \geq 0$ for every nonnegative $φ$ ; hence both profiles satisfy all Kruzhkov entropy inequalities.

4.1step 1.1step 2.1step 2.2step 1.2step 2.3step 3.1∎

The constant case and uniqueness. If $u_{L} = u_{R}$ , the constant $u \equiv u_{L}$ is a distributional weak solution with the exact trace, and its entropy production vanishes, so it is a Kruzhkov entropy solution; any bounded Kruzhkov entropy solution with the same constant datum equals it by order preservation applied in both directions. In the cases (i) and (ii), the profiles are bounded Kruzhkov entropy solutions with datum $u_{0}$ by steps 1.1, 2.1, 2.2–2.3 and 3.1, and any bounded Kruzhkov entropy solution with datum $u_{0}$ coincides with the profile almost everywhere on $Π_{T}$ by Uniqueness, comparison and order preservation of entropy solutions applied in both directions. This proves existence, uniqueness, and the asserted profile in each case.

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Oleinik's one-sided estimate characterizes bounded entropy solutions

Statement

Assume Countable Choice and Dependent Choice, as used by the mollification and vanishing-viscosity compactness interfaces. Let $T > 0$ , $f \in C^{2} (R)$ , and let $I \subset R$ be a bounded closed interval with $f^{''} \geq κ > 0$ on $I$ . Let $u_{0} \in L^{\infty} (R)$ and let $u \in L^{\infty} ((0, T) \times R)$ be a distributional weak solution of $u_{t} + \partial_{x} f (u) = 0$ , with $u_{0}$ and the essential range of $u$ in $I$ , and with the strong local $L^{1}$ initial trace $u_{0}$ in Kruzhkov entropy solutions. The following are equivalent: (i) $u$ is a Kruzhkov entropy solution with that trace; (ii) for almost every $t \in (0, T)$ and almost every pair $x < y$ , $u (t, y) - u (t, x) \leq \frac{y - x}{κ t};$ (iii) for almost every $t \in (0, T)$ , $D_{x} u (t, \cdot) \leq (κ t)^{- 1} L^{1}$ in distributional order. The initial trace and weak equation are hypotheses of the equivalence; the slope bound alone is not a definition of an entropy solution. For piecewise $C^{1}$ solutions, the bound in particular excludes upward jumps, and the convex chord criterion makes the remaining shocks entropy-admissible. The state-slope constant $1 / κ$ requires uniform convexity on the solution range.

Facts & Assumptions

Given: Countable and Dependent Choice, $T > 0$ , $f \in C^{2} (R)$ , a bounded closed interval $I$ with $f^{''} \geq κ > 0$ on $I$ , a bounded weak solution $u$ with datum $u_{0} \in L^{\infty}$ , both taking values in $I$ , and a nonnegative test function $φ$ in the arguments below.

[F1]

The weak equation and the trace: $\int_{Π_{T}} (u φ_{t} + f (u) φ_{x}) = 0$ for every $φ \in C_{c}^{\infty} (Π_{T})$ , and $u$ has the strong local $L^{1}$ trace $u_{0}$ ; a Kruzhkov entropy solution is a bounded weak solution satisfying $\partial_{t} ∣ u - k ∣ + \partial_{x} q_{k} (u) \leq 0$ for all $k$ , $q_{k} (s) = sgn (s - k) (f (s) - f (k))$ (Kruzhkov entropy solutions).

[F2]

The viscous construction supplies bounded $C^{1, 2}$ solutions for $C^{2}$ fluxes and smooth compactly supported data (The viscous scalar Cauchy problem with smooth data has a global classical solution). Its heat-potential cancellation estimates apply on every positive-time strip (The heat evolution $H_{t}$ of initial data, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel). For a smooth flux, these also make $p = v_{x}$ classical: $p$ satisfies the differentiated divergence equation with source $g^{'} (v) p$ , which is parabolically Hölder by the gradient estimate in the construction. Applying its second-kernel cancellation first gives Hölder $p_{x}$ ; then the source $- g^{'} (v) p_{x} - g^{''} (v) p^{2}$ is Hölder, and the nondifferentiated heat-potential estimate gives $p \in C^{1, 2}$ locally. Chain and product rules are The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ . Smooth flux regularization is used before this differentiation.

[F3]

Vanishing viscosity: for each smooth compactly supported datum the viscous solutions have a subsequence converging in $L_{l o c}^{1} (Π_{T})$ and almost everywhere to the unique bounded Kruzhkov entropy solution of that datum, and the entropy inequalities pass to the limit (Existence of bounded Kruzhkov entropy solutions, Vanishing-viscosity families are locally precompact in $L^{1}$ ).

[F4]

Local contraction: two bounded Kruzhkov entropy solutions with data in $L_{l o c}^{1} \cap L^{\infty}$ satisfy $\int_{B (x_{0}, R - L^{'} t)} ∣ v - w ∣ \leq \int_{B (x_{0}, R)} ∣ v_{0} - w_{0} ∣$ for almost every $t$ with $L^{'} t < R$ , where $L^{'}$ is a Lipschitz constant of the (shifted) flux on the common range (Local $L^{1}$ contraction for two entropy solutions).

[F5]

Mollification and distributional calculus: convolutions with radial mollifiers are smooth and converge in $L^{1}$ (or $L_{l o c}^{1}$ ) to the original function; derivatives may be taken inside the convolution; approximate identities converge in $L^{p}$ ; distributional derivatives commute with convolution against test functions; almost-everywhere convergence is available along subsequences of $L^{1}$ -convergent sequences (A radial mollifier family in Rn, A unit-mass smooth bump generates an $L^{1}$ approximate identity, Convolution with a mollifier is smooth, and derivatives pass under the integral sign, Distributional derivatives commute with test-function convolution, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , Convergence in L^1(mu) has an almost-everywhere convergent subsequence, Dominated convergence, Fubini's theorem for L^1 functions on a sigma-finite product, The space $L^{p} (μ)$ as the quotient by null functions).

[F6]

Piecewise $C^{1}$ interpretation: at a single shock satisfying the Rankine--Hugoniot condition, the entropy inequalities for all convex $C^{2}$ pairs hold iff the chord residual $F (z) = f (z) - f (u^{-}) - s (z - u^{-})$ satisfies $F (z) (u^{+} - u^{-}) \geq 0$ between the states; in particular only compressive jumps $u^{-} > u^{+}$ are admissible (The convex entropy condition for a single shock is the chord condition).

Proof

technique · direct

1.1F2

Shifting and positive-strip gradient bounds. Fix $c \in I$ and put $J : = I - c$ , which contains $0$ . Set $\tilde{f} (s) : = f (s) - f (0)$ and $g (v) : = \tilde{f} (v + c) - \tilde{f} (c) = f (v + c) - f (c) .$ Then $g (0) = 0$ and $g^{''} \geq κ$ on $J$ . For a smooth compactly supported datum $v_{0}^{s}$ with values in $J$ , apply [F2] to flux $g$ and datum $v_{0}^{s}$ , obtaining $v^{ε}$ with values in $J$ . The unshifted profile $U^{ε} : = v^{ε} + c$ has datum $v_{0}^{s} + c$ , equals $c$ outside a compact set, and solves the equation with normalized flux $\tilde{f}$ ; thus the compact datum to which [F2] is applied is $v_{0}^{s}$ , not the generally noncompact function $v_{0}^{s} + c$ or a nonzero-tail shift. The Oleinik slope is unchanged by adding $c$ . Set $M_{v} : = ∥ v_{0}^{s} ∥_{\infty}$ and $G_{0} : = \sup_{v \in J} ∣ g^{'} (v) ∣ < \infty$ . By variation of constants, for $0 < d < t$ , $v^{ε} (t) = H_{ε (t - d)} v^{ε} (d) - \int_{d}^{t} \partial_{x} H_{ε (t - s)} g (v^{ε} (s)) d s .$ Indeed, differentiating $H_{ε (t - s)} v^{ε} (s)$ for $d < s < t$ and using the viscous equation gives $- \partial_{x} H_{ε (t - s)} g (v^{ε} (s))$ ; the integral converges at $s = t$ because $∥ \partial_{x} Γ_{ε r} ∥_{1} \leq C_{ε} r^{- 1 / 2}$ . The Gaussian derivative bounds [F2] also give $∥ \partial_{x} Γ_{ε r} (\cdot + h) - \partial_{x} Γ_{ε r} ∥_{1} \leq \min (C_{ε} r^{- 1 / 2}, C_{ε} ∣ h ∣ r^{- 1}), ∥ \partial_{r} \partial_{x} Γ_{ε r} ∥_{1} \leq C_{ε} r^{- 3 / 2} .$ Splitting the spatial integral at $r = ∣ h ∣^{2}$ and the time integral at $r = ∣ t - t^{'} ∣$ in the restarted identity shows that, for every $0 < α < 1$ and every $τ > 0$ , $∣ v^{ε} (t, x + h) - v^{ε} (t, x) ∣ \leq C_{τ, T} ∣ h ∣^{α}, ∣ v^{ε} (t, x) - v^{ε} (t^{'}, x) ∣ \leq C_{τ, T} ∣ t - t^{'} ∣^{1 / 2}$ on $[τ / 2, T] \times R$ . Fix $τ > 0$ , put $d = τ / 2$ , and differentiate the restarted identity for $t \geq τ$ , first omitting its final $η$ -length of integration and then letting $η ↓ 0$ . Since $\int \partial_{x x} Γ_{ε r} = 0$ , cancellation gives $p (t, x) = \partial_{x} H_{ε (t - d)} v^{ε} (d, x) - \int_{0}^{t - d} \int_{R} \partial_{x x} Γ_{ε r} (z) (g (v^{ε} (t - r, x - z)) - g (v^{ε} (t, x))) d z d r .$ The range bound and the positive-strip estimates imply $∣ g (v^{ε} (t - r, x - z)) - g (v^{ε} (t, x)) ∣ \leq G_{0} C_{τ, T} (∣ z ∣^{α} + r^{1 / 2})$ . By scaling, $\int ∣ \partial_{x x} Γ_{ε r} (z) ∣ (∣ z ∣^{α} + r^{1 / 2}) d z \leq C_{ε, α} (r^{- 1 + α / 2} + r^{- 1 / 2}),$ which is integrable at $r = 0$ ; the first term is bounded by $C_{ε, τ} M_{v}$ because $t - d \geq τ / 2$ . Thus $\sup_{(t, x) \in [τ, T] \times R} ∣ p (t, x) ∣ < \infty .$ This bound is global in space and holds on every positive-time strip.

1.2F5

Equivalence of the pointwise and distributional forms. Fix $t \in (0, T)$ , assume first that (ii) holds at this $t$ , and put $w (x) = u (t, x) - x / (κ t)$ , so that $w (y) \leq w (x)$ for almost every $x < y$ . For $n \geq 1$ set $a : = 1 / n$ and $w_{n} (x) = n \int_{x}^{x + a} w (z) d z$ . If $x < x^{'}$ , put $d : = x^{'} - x > 0$ . When $d \leq a$ , cancellation of the overlap gives $w_{n} (x) - w_{n} (x^{'}) = n (\int_{x}^{x + d} w (z) d z - \int_{x + a}^{x + a + d} w (ζ) d ζ) \geq 0,$ because the two intervals have equal length $d$ and every interior pair $z \in [x, x + d]$ , $ζ \in [x + a, x + a + d]$ satisfies $z < ζ$ ; explicitly, $d (\int_{x}^{x + d} w - \int_{x + a}^{x + a + d} w) = \int_{[x, x + d] \times [x + a, x + a + d]} (w (z) - w (ζ)) d z d ζ \geq 0$ by the assumed a.e. pair inequality. When $d \geq a$ , the intervals $[x, x + a]$ and $[x^{'}, x^{'} + a]$ are ordered and have equal length, so $w_{n} (x) - w_{n} (x^{'}) = n (\int_{x}^{x + a} w (z) d z - \int_{x^{'}}^{x^{'} + a} w (ζ) d ζ) \geq 0,$ since $a (\int_{x}^{x + a} w - \int_{x^{'}}^{x^{'} + a} w) = \int_{[x, x + a] \times [x^{'}, x^{'} + a]} (w (z) - w (ζ)) d z d ζ \geq 0$ . Thus $w_{n}$ is nonincreasing and $D_{x} w_{n} \leq 0$ distributionally. Since $w_{n} \to w$ in $L_{l o c}^{1}$ as $n \to \infty$ , distributional differentiation passes to the limit, so $D_{x} w \leq 0$ , which is (iii) at this $t$ . Conversely, assume (iii) at some $t$ and let $ρ_{δ}$ be a nonnegative spatial mollifier; then $w_{δ} = w * ρ_{δ}$ is smooth with $D_{x} w_{δ} = (D_{x} w) * ρ_{δ} \leq 0$ , so $w_{δ}$ is nonincreasing and $w_{δ} (y) \leq w_{δ} (x)$ for all $x < y$ ; since $w_{δ} \to w$ in $L_{l o c}^{1}$ , passing to an almost-everywhere convergent subsequence gives the two-point inequality of (ii) at this $t$ for almost every pair. Hence (ii) and (iii) hold for the same full-measure set of times, proving the equivalence.

1.3F5

Mollification commutator. Assume (iii). Fix a nonnegative test function $φ$ supported in $[a, b] \times [- R, R]$ with $0 < a < b < T$ , and let $ρ_{δ}$ be a nonnegative unit-mass mollifier on $R^{2}$ supported in the ball of radius $δ < \frac{1}{2} \min {a, T - b}$ . Extend $u$ boundedly to all of $R^{2}$ by a fixed value in $I$ outside $(0, T) \times R$ , and set $u_{δ} = u * ρ_{δ}$ , $F_{δ} = f (u) * ρ_{δ}$ , $r_{δ} = f (u_{δ}) - F_{δ}$ . By [F5], distributional derivatives commute with convolution, so on a neighbourhood of the support of $φ$ the identity $\partial_{t} u_{δ} + \partial_{x} F_{δ} = 0$ holds; since the mollification averages only over times $\geq a - δ > a / 2$ , the distributional bound (iii) gives $\partial_{x} u_{δ} \leq 2 / (κ a)$ there; and $u_{δ}$ still takes values in $I$ . The tangent inequality $f (s) \geq f (u_{δ}) + f^{'} (u_{δ}) (s - u_{δ})$ for convex $f$ , averaged against $ρ_{δ}$ , gives $r_{δ} \leq 0$ . Finally, localizing $u$ and $f (u)$ by a cutoff equal to $1$ on a slightly larger compact set and applying approximate-identity convergence in $L^{1}$ [F5] gives $u_{δ} \to u$ , $F_{δ} \to f (u)$ and $r_{δ} \to 0$ in $L^{1}$ on the support of $φ$ , the last two also using that $f$ is Lipschitz on the bounded interval $I$ .

2.1F2F5step 1.1

A classical barrier after flux regularization. Take smooth normalized $g_{l}$ converging to $g$ in $C^{2}$ on the compact state interval $J$ , with $g_{l} (0) = 0$ and $g_{l}^{''} \geq κ_{l} : = κ - 1 / l > 0$ there (mollify $g$ at sufficiently small scales). Choose $0 < ε_{l} \leq 1$ tending to zero, and let $v^{l}$ have the fixed smooth datum of step 1.1 and flux $g_{l}$ . Its range lies in $J$ . By [F2], $p = v_{x}^{l}$ is classical at positive times, is bounded globally on positive strips by step 1.1, and satisfies $p_{t} + g_{l}^{'} (v^{l}) p_{x} + g_{l}^{''} (v^{l}) p^{2} = ε_{l} p_{x x}$ . Fix $τ > 0$ , put $q (t) = 1 / (κ_{l} (t - τ))$ , $b = g_{l}^{'} (v^{l})$ and $W = p - q$ . Where $W > 0$ , $(\partial_{t} + b \partial_{x} - ε_{l} \partial_{x x}) W < 0$ since $p > q > 0$ . If $G_{l} = \sup_{J} ∣ g_{l}^{'} ∣$ , choose $A > G_{l} + 2 ε_{l}$ and $Φ = e^{A (t - τ)} (1 + x^{2})$ , so the same operator applied to $Φ$ is strictly positive. For $ρ > 0$ , $Z = W - ρ Φ$ is negative at some $t_{0} > τ$ sufficiently close to $τ$ , by the positive-strip bound on $p$ , and negative on the sides of a sufficiently large rectangle. A positive maximum on that rectangle would have $Z_{t} \geq 0$ , $Z_{x} = 0$ , $Z_{x x} \leq 0$ , contradicting the strict operator inequality there. Thus $Z \leq 0$ . Let $ρ ↓ 0$ and then $τ ↓ 0$ to get $v_{x}^{l} \leq 1 / (κ_{l} t)$ ; integrating in $x$ gives $v^{l} (t, y) - v^{l} (t, x) \leq (y - x) / (κ_{l} t)$ . This uses a classical maximum argument, with no Sobolev positive-part test.

2.2F5step 1.3

The entropy production tends to a nonpositive limit. For any convex $η \in C^{2}$ and $q^{'} = η^{'} f^{'}$ , the chain and product rules applied to $\partial_{t} u_{δ} + \partial_{x} (f (u_{δ}) - r_{δ}) = 0$ give $\partial_{t} η (u_{δ}) + \partial_{x} q (u_{δ}) = \partial_{x} (η^{'} (u_{δ}) r_{δ}) - η^{''} (u_{δ}) (\partial_{x} u_{δ}) r_{δ} \leq \partial_{x} (η^{'} (u_{δ}) r_{δ}) + \frac{2 ∥ η^{''} ∥_{L^{\infty} (I)}}{κ a} (- r_{δ})$ , using $r_{δ} \leq 0$ and $\partial_{x} u_{δ} \leq 2 / (κ a)$ from step 1.3. Tested against the nonnegative $φ$ , the first term is bounded by $∥ η^{'} ∥_{L^{\infty} (I)} ∥ φ_{x} ∥_{\infty} ∥ r_{δ} ∥_{L^{1} (supp φ)}$ after an integration by parts, and the second by $\frac{2 ∥ η^{''} ∥_{\infty}}{κ a} ∥ φ ∥_{\infty} ∥ r_{δ} ∥_{L^{1} (supp φ)}$ ; both tend to $0$ as $δ ↓ 0$ . Since $\partial_{t} η (u_{δ}) + \partial_{x} q (u_{δ}) \to \partial_{t} η (u) + \partial_{x} q (u)$ distributionally by [F5] (local $L^{1}$ convergence of $u_{δ}$ and continuity of $η, q$ ), the limit satisfies $⟨ \partial_{t} η (u) + \partial_{x} q (u), φ ⟩ \leq 0$ for every nonnegative test function supported in $(0, T) \times R$ .

3.1F3F4F5step 2.1

Passage to the smooth-datum entropy solution. The varying-flux family $v^{l}$ of step 2.1 satisfies the common range and derivative bounds of the compactness lemma [F3]. Extract a locally $L^{1}$ and almost-everywhere convergent subsequence. The existence proof passes its weak and entropy identities to the limit because $g_{l} \to g$ in $C^{1}$ on the range; the uniform local time modulus supplies the initial trace. Uniqueness identifies the limit as the entropy solution $v$ for $g$ with this smooth datum. Fubini gives slicewise convergence at almost every time, and passage to the bound in step 2.1, with $κ_{l} \to κ$ , gives $v (t, y) - v (t, x) \leq (y - x) / (κ t)$ for almost every time and almost every pair $x < y$ .

4.1F4F5step 3.1

Approximation for general data: (i) implies (ii). Now let $u$ be the given entropy solution with datum $u_{0}$ and range in $I$ , and fix $c \in I$ . For $j \geq 1$ set $w_{j} = ((u_{0} - c) 1_{[- j, j]}) * ρ_{δ_{j}}$ , where $ρ_{δ_{j}}$ is a nonnegative mollifier of radius $δ_{j} ↓ 0$ : then $w_{j} \in C_{c}^{\infty}$ , its values lie in $I - c$ (a convex combination of values of $u_{0} - c$ ), and $w_{j} \to u_{0} - c$ in $L_{l o c}^{1} (R)$ . Let $v_{j}$ be the entropy solution with datum $w_{j}$ for the flux $g$ : by step 3.1 each $v_{j}$ satisfies the two-point estimate, and by [F4] applied to $v_{j}$ and $\tilde{u} = u - c$ , the differences converge to $0$ in $L^{1}$ on every compact cylinder; a diagonal subsequence converges almost everywhere on $Π_{T}$ . Passing the two-point estimate to that almost-everywhere limit proves (ii) for $u$ .

5.1F1F5step 4.1step 1.2step 2.2

Kruzhkov pairs and conclusion of (iii) implies (i). For $k \in R$ and $m \geq 1$ put $η_{m} (s) = \sqrt{(s - k)^{2} + m^{- 2}}$ and $q_{m} (s) = \int_{k}^{s} η_{m}^{'} (z) f^{'} (z) d z$ , so $(η_{m}, q_{m})$ is a smooth convex entropy pair. Step 2.2 gives $\partial_{t} η_{m} (u) + \partial_{x} q_{m} (u) \leq 0$ for every $m$ ; letting $m \to \infty$ , $η_{m} \to ∣ s - k ∣$ and $q_{m} \to sgn (s - k) (f (s) - f (k))$ uniformly on the bounded interval $I$ by dominated convergence, so the distributional inequality passes to the limit and every Kruzhkov inequality holds. Together with the weak equation and the strong local $L^{1}$ trace (hypotheses), $u$ is a Kruzhkov entropy solution, proving (iii) implies (i); the reverse implication (i) implies (ii) is step 4.1, and the equivalence of (ii) and (iii) is step 1.2.

6.1F6step 4.1step 1.2step 5.1∎

The piecewise $C^{1}$ remark. If $u$ is piecewise $C^{1}$ with a single jump at a curve $x = s (t)$ and satisfies the hypotheses, then (ii) forces the right trace not to exceed the left trace across an upward jump: taking $x ↑ s (t)$ , $y ↓ s (t)$ in the two-point inequality and letting $x, y \to s (t)$ gives $u^{+} - u^{-} \leq 0$ , so an upward jump $u^{+} > u^{-}$ is excluded; for the remaining jumps with $u^{-} > u^{+}$ the Rankine--Hugoniot condition and the chord criterion [F6] make them entropy-admissible. This shows how the slope bound encodes admissibility in the piecewise smooth class, while the equivalence itself was proved for all bounded weak solutions.

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The Lax shock inequalities for convex scalar laws

Statement

Let $f \in C^{2} (R)$ be strictly convex. Consider a nontrivial one-dimensional jump from the left trace $u^{-}$ to the right trace $u^{+}$ across $x = s (t)$ , with traces as in Piecewise smooth shocks and one-sided traces, satisfying the Rankine--Hugoniot condition of The Rankine--Hugoniot jump condition in space--time normal form. The jump is Kruzhkov entropy-admissible (Kruzhkov entropy solutions) if and only if $u^{-} > u^{+}$ . In that case its speed is $s^{'} = \frac{f (u^{+}) - f (u^{-})}{u^{+} - u^{-}},$ and it satisfies the Lax shock inequalities $f^{'} (u^{+}) \leq s^{'} \leq f^{'} (u^{-}) .$ In particular, every nontrivial entropy-admissible jump is compressive; no admissible jump increases the state across the shock (Convex and strictly convex functions on Euclidean convex sets).

Facts & Assumptions

Given: a strictly convex $f \in C^{2} (R)$ , a nontrivial single-jump piecewise $C^{1}$ weak solution with left trace $u^{-}$ , right trace $u^{+}$ across $x = s (t)$ , and speed $σ = s^{'}$ satisfying the Rankine--Hugoniot condition.

[F1]

Rankine--Hugoniot and jump setup: the interface is the graph $x = s (t)$ with minus side $x < s (t)$ and plus side $x > s (t)$ , and $σ (u^{+} - u^{-}) = f (u^{+}) - f (u^{-})$ , i.e. $σ = (f (u^{+}) - f (u^{-})) / (u^{+} - u^{-})$ since the jump is nontrivial (Piecewise smooth shocks and one-sided traces, The Rankine--Hugoniot jump condition in space--time normal form).

[F2]

Chord criterion: with $F (z) = f (z) - f (u^{-}) - σ (z - u^{-})$ , the jump satisfies the Kruzhkov entropy inequalities for all convex entropy pairs if and only if $F (z) (u^{+} - u^{-}) \geq 0$ for every $z$ between $u^{-}$ and $u^{+}$ ; this is the notion of entropy admissibility at a single jump (The convex entropy condition for a single shock is the chord condition, Kruzhkov entropy solutions).

[F3]

Strict convexity: for a differentiable strictly convex $f$ , the graph lies strictly below every chord on the interior of its interval; the derivative $f^{'}$ is strictly increasing: it is nondecreasing by the cited theorem, and equality at $a < b$ would make it constant on $[a, b]$ , so FTC would make $f$ affine there, contradicting strict convexity; and for $a < b$ the difference quotients satisfy $f^{'} (a) \leq \frac{f (b) - f (a)}{b - a} \leq f^{'} (b)$ with strict inequalities throughout, while the mean value theorem gives $\frac{f (b) - f (a)}{b - a} = f^{'} (c)$ for some $c \in (a, b)$ (Convex and strictly convex functions on Euclidean convex sets, A differentiable function on an open interval is convex if and only if its derivative is nondecreasing, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

Proof

technique · direct

1.1F1F2F3

The forward case is never admissible. Suppose $u^{-} < u^{+}$ . The chord criterion of [F2] requires $F (z) \geq 0$ for all $z \in [u^{-}, u^{+}]$ . By strict convexity [F3] the graph of $f$ lies strictly below the chord through $(u^{-}, f (u^{-}))$ and $(u^{+}, f (u^{+}))$ on $(u^{-}, u^{+})$ , and that chord is $z \mapsto f (u^{-}) + σ (z - u^{-})$ because its slope is $σ$ ; hence $F (z) < 0$ for every $z \in (u^{-}, u^{+})$ , contradicting the criterion. So a nontrivial Rankine--Hugoniot jump with $u^{-} < u^{+}$ is not entropy-admissible.

2.1F1F2F3

The backward case is admissible. Suppose $u^{-} > u^{+}$ . On the interval between the states, strict convexity gives $F (z) < 0$ for $u^{+} < z < u^{-}$ and $F (u^{+}) = F (u^{-}) = 0$ . Since $u^{+} - u^{-} < 0$ , the product $F (z) (u^{+} - u^{-})$ is positive for interior $z$ and vanishes at the endpoints, so the chord criterion of [F2] holds and the jump is entropy-admissible. Together with step 1.1 this shows that a nontrivial Rankine--Hugoniot jump is entropy-admissible if and only if $u^{-} > u^{+}$ ; in particular no admissible jump increases the state.

3.1F1F3step 1.1step 2.1∎

The Lax inequalities. Assume $u^{-} > u^{+}$ and write $a = u^{+} < b = u^{-}$ . By [F1], $σ = \frac{f (b) - f (a)}{b - a} = \frac{f (u^{+}) - f (u^{-})}{u^{+} - u^{-}}$ , which is the stated speed. By the mean value theorem [F3] there is $c \in (a, b)$ with $f^{'} (c) = σ$ , and since $f^{'}$ is strictly increasing, $f^{'} (u^{+}) = f^{'} (a) < f^{'} (c) = σ < f^{'} (b) = f^{'} (u^{-})$ ; a fortiori $f^{'} (u^{+}) \leq σ \leq f^{'} (u^{-})$ , the Lax shock inequalities.

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The Hamilton--Jacobi correspondence in one dimension

Statement

(i) Let $U_{0} \in C^{0, 1} (R) \cap L^{\infty} (R)$ and let $u_{0} = U_{0}^{'}$ be its a.e. derivative. The Hopf--Lax function $V (t, x) = Q_{t} U_{0} (x) = \inf_{y \in R} {U_{0} (y) + t L ⁣ (\frac{x - y}{t})}, t > 0, V (0, x) = U_{0} (x),$ where $L = f^{*}$ , is the unique viscosity solution of $V_{t} + f (V_{x}) = 0$ with initial datum $U_{0}$ among functions bounded and uniformly continuous on $[0, S] \times R$ for every finite $S > 0$ (The Hamilton--Jacobi Cauchy problem and its classical solutions, Discontinuous viscosity solutions through the two envelopes). Its a.e. spatial derivative $v = V_{x}$ is the bounded Kruzhkov entropy solution of $v_{t} + \partial_{x} f (v) = 0$ with initial datum $u_{0}$ .

(ii) Conversely, let $u_{0} \in L^{1} (R) \cap L^{\infty} (R)$ have compact support and let $u$ be its bounded Kruzhkov entropy solution, using the strong local $L^{1}$ initial trace. With $U_{0} (x) = \int_{- \infty}^{x} u_{0} (y) d y, U (t, x) = \int_{- \infty}^{x} u (t, y) d y - t f (0),$ the function $U$ is the unique viscosity solution of $U_{t} + f (U_{x}) = 0$ with datum $U_{0}$ in the same finite-slab class as in (i), and $U_{x} = u$ almost everywhere. In the compactly supported datum class of (ii), differentiation and the normalized primitive are inverse correspondences (Kruzhkov entropy solutions, Existence of bounded Kruzhkov entropy solutions).

Facts & Assumptions

Given: Countable and Dependent Choice, a strictly convex superlinear $C^{2}$ flux $f$ , its conjugate $L$ , and the two datum classes in the statement.

[F1]

Hopf--Lax is a viscosity solution bounded and uniformly continuous on each finite time slab for bounded uniformly continuous data, unique in that finite-slab class; its minimisers exist, it has the semigroup property, and it contracts the supremum norm (The Hopf--Lax operator and the Hopf--Lax formula, Finiteness, superlinearity of the Lagrangian and localisation of Hopf--Lax near-minimisers, The Hopf--Lax formula solves the Hamilton--Jacobi Cauchy problem, The Hopf--Lax operator is a contraction in the supremum norm, The Legendre transform of a finite-valued convex Hamiltonian, The Hamilton--Jacobi Cauchy problem and its classical solutions, Discontinuous viscosity solutions through the two envelopes, Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem).

[F2]

The normalized flux $g = f - f (0)$ gives the same conservation law. For smooth compactly supported data its viscous solutions are mild classical solutions, obey the range and $L^{1}$ bounds, and are locally precompact in space--time $L^{1}$ , with limits continuous into local $L^{1}$ . The existence proof passes their weak and entropy identities to the unique entropy solution (The viscous scalar Cauchy problem with smooth data has a global classical solution, Uniform L-infinity, mass and energy bounds for the viscous approximations, Vanishing-viscosity families are locally precompact in $L^{1}$ , Existence of bounded Kruzhkov entropy solutions, Kruzhkov entropy solutions, Distributional weak solutions of the Cauchy problem).

[F3]

The heat kernels have unit mass, solve the heat equation, have Gaussian derivative estimates, and give the heat evolution; smooth cutoffs have derivatives $O (R^{- 1})$ and $O (R^{- 2})$ (The heat evolution $H_{t}$ of initial data, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel, Explicit compactly supported smooth cutoffs). Fubini, dominated convergence and FTC justify the kernel calculations (Fubini's theorem for L^1 functions on a sigma-finite product, Dominated convergence, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F4]

Entropy solutions contract in global $L^{1}$ at every time for their continuous representatives, and locally on shrinking balls; they are unique in the bounded class. Compactly supported data stay supported in a common bounded interval on every finite horizon (Global $L^{1}$ contraction from the local estimate, Local $L^{1}$ contraction for two entropy solutions, Uniqueness, comparison and order preservation of entropy solutions, Finite propagation for scalar conservation laws).

[F5]

Under the declared choice assumptions, indefinite integrals of $L^{1}$ functions are absolutely continuous and differentiate to their integrands almost everywhere (The indefinite integral of an $L^{1}$ function is absolutely continuous, The indefinite integral of an $L^{1}$ function is differentiable almost everywhere). Lipschitz functions are absolutely continuous on compact intervals, so their a.e. derivatives recover their increments by Fundamental theorem of calculus for absolutely continuous functions. $L^{1}$ is complete. Mollification gives smooth compactly supported approximations to compactly supported bounded data in $L^{1}$ , with the same bound, with norm convergence supplied by Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ (Fundamental theorem of calculus for absolutely continuous functions, Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , An $L^{1}$ approximate identity on $R^{n}$ , A unit-mass smooth bump generates an $L^{1}$ approximate identity, Convolution with a mollifier is smooth, and derivatives pass under the integral sign, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct

1.1F1

Conjugate calculus and localization. Strict convexity makes $f^{'}$ strictly increasing: convex secant inequalities give monotonicity, and equality at two distinct points would make $f$ affine between them. Superlinearity makes its limits $\pm \infty$ (a finite derivative bound at either end would bound $f$ linearly there). Thus $p (q) : = (f^{'})^{- 1} (q)$ is continuous, and is the unique maximiser of $p q - f (p)$ . The inequalities $p (q) h \leq L (q + h) - L (q) \leq p (q + h) h$ for $h > 0$ , and their reversed versions for $h < 0$ , show that $L^{'} (q) = p (q)$ . If $W$ is $M$ -Lipschitz and $y$ minimises $W (y) + t L ((x - y) / t)$ , perturb $y$ in each direction and use the Lipschitz bound to obtain $∣ L^{'} ((x - y) / t) ∣ \leq M$ . Hence $∣ x - y ∣ \leq C t$ , where $C = \max_{∣ p ∣ \leq M} ∣ f^{'} (p) ∣$ . Translating competitors shows that $Q_{t} W$ is $M$ -Lipschitz in $x$ . Moreover $L (v) - M ∣ v ∣ \geq - C_{0}$ , where $C_{0} = \max {f (M), f (- M)}$ by the conjugate definition, and the competitor $y = x$ gives $Q_{h} W - W \leq h L (0)$ . These bounds and the semigroup law give a uniform time Lipschitz bound on finite horizons. Also $L (q) \geq - f (0)$ for all $q$ , with equality at $q = f^{'} (0)$ , so $Q_{t} 0 = - t f (0)$ . Sup contraction therefore gives $∥ Q_{t} U_{0} + t f (0) ∥_{\infty} \leq ∥ U_{0} ∥_{\infty}$ , proving boundedness on each finite slab, with no time-uniform bound asserted.

1.2F2F3

Viscous primitives for a smooth compact datum. Let $a \in C_{c}^{\infty}$ , $P_{0} (x) = \int_{- \infty}^{x} a$ , and let $u^{ε}$ be [F2]'s viscous solution for $g = f - f (0)$ . Define $W^{ε} (t) = H_{ε t} P_{0} - \int_{0}^{t} H_{ε (t - s)} f (u^{ε} (s)) d s .$ Differentiating in $x$ gives exactly the mild identity for $u^{ε}$ , so $W_{x}^{ε} = u^{ε}$ . Heat-potential cancellation as in the viscous construction makes $W^{ε}$ classical at positive times, and its equation is $W_{t}^{ε} + f (W_{x}^{ε}) = ε W_{x x}^{ε}$ . At $x \to - \infty$ its initial heat term tends to zero, while the integral tends to $t f (0)$ : $g (u^{ε} (s)) \in L^{1}$ , and convolution of an $L^{1}$ function with a bounded Gaussian tends to zero at spatial infinity; boundedness dominates the finite time integral. Thus $W^{ε} (t, x) = \int_{- \infty}^{x} u^{ε} (t, y) d y - t f (0)$ . Testing the smoothed $∣ u^{ε} ∣$ balance with an exterior cutoff, then removing a second outer cutoff, gives $\int ∣ u^{ε} (t) ∣ (1 - χ_{R}) \leq \int ∣ a ∣ (1 - χ_{R}) + C T (R^{- 1} + R^{- 2}) ∥ a ∥_{1} (0 \leq t \leq T, 0 < ε \leq 1) .$ This is the cutoff calculation of the $L^{1}$ bound in [F2], with $∣ g (u) ∣ \leq C ∣ u ∣$ and [F3]'s derivative bounds. It supplies uniform tails.

2.1F2F3step 1.2

A time modulus for the viscous primitives. The Gaussian convolution identity $H_{r} H_{s} = H_{r + s}$ follows by completing the square in the kernel product and using unit mass and Fubini. Applying it to the definition in step 1.2 gives $W^{ε} (t + h) = H_{ε h} W^{ε} (t) - \int_{t}^{t + h} H_{ε (t + h - s)} f (u^{ε} (s)) d s$ . Put $M = ∥ a ∥_{\infty}$ and $B = \max_{∣ z ∣ \leq M} ∣ f (z) ∣$ . Since $W_{x}^{ε} = u^{ε}$ , these primitives are $M$ -Lipschitz in $x$ , including at $t = 0$ . Gaussian scaling gives $\int ∣ z ∣ Γ (z, r) d z = C \sqrt{r}$ , with $C < \infty$ by the Gaussian bound. Hence unit mass gives $∥ H_{r} W^{ε} (t) - W^{ε} (t) ∥_{\infty} \leq C M \sqrt{r}$ , and heat contraction bounds the time integral by $B h$ . Thus $∥ W^{ε} (t + h) - W^{ε} (t) ∥_{\infty} \leq C M \sqrt{h} + B h$ for $0 \leq t < t + h \leq T$ , uniformly in $0 < ε \leq 1$ .

3.1F2F3step 1.2step 2.1

Uniform convergence of primitives. By [F2], choose a subsequence $u^{ε} \to u$ locally in space--time $L^{1}$ , where $u$ is the entropy solution of datum $a$ , continuous into local $L^{1}$ . A further subsequence converges on almost every time slice locally in $L^{1}$ . The uniform tails of step 1.2 pass to these slices by monotone exhaustion, and to every time by local continuity on bounded annuli followed by exhaustion. They imply $u (t) \in L^{1}$ with uniformly small tails; local continuity then gives global $L^{1}$ continuity on $[0, T]$ . Moreover $\int_{0}^{T} ∥ u^{ε} (t) - u (t) ∥_{1} d t \to 0$ : the tails are uniformly small outside large intervals, the compact space--time convergence handles times away from $0, T$ , and the bound $∣ u^{ε} ∣, ∣ u ∣ \leq M$ controls the remaining small time intervals on the fixed spatial interval. For $P (t, x) = \int_{- \infty}^{x} u (t, y) d y - t f (0)$ , the primitive formula of step 1.2 gives $∥ W^{ε} (t) - P (t) ∥_{\infty} \leq ∥ u^{ε} (t) - u (t) ∥_{1}$ , so the integral in time of the left side tends to zero. The function $P$ is continuous in time in the supremum norm by global $L^{1}$ continuity. Together with the common modulus of step 2.1, this implies uniform convergence on $[0, T] \times R$ : a discrepancy of size $d > 0$ at any time would persist with size at least $d / 2$ on a one-sided interval of length bounded below independently of $ε$ , contradicting that vanishing time integral.

4.1F1F2F3step 1.2step 3.1

The viscosity limit. At a strict local maximum of $P - ϕ$ , with smooth $ϕ$ , step 3.1 gives nearby local maxima of $W^{ε} - ϕ$ . The classical equation in step 1.2 gives $ϕ_{t} + f (ϕ_{x}) \leq ε ϕ_{x x}$ there; passing to the limit proves the subsolution inequality. Local minima give the supersolution inequality. Adding a fourth-power distance term makes a contact strict without changing its first derivatives; approximation in $C^{1}$ on a compact contact neighbourhood reduces $C^{1}$ tests to smooth tests. Thus $P$ is a viscosity solution with initial datum $P_{0}$ . It is bounded by $∥ a ∥_{1} + T ∣ f (0) ∣$ , spatially $M$ -Lipschitz, and uniformly continuous in time on $[0, T]$ by step 3.1. Uniqueness in [F1] gives $P = Q_{t} P_{0}$ .

5.1F1F4F5step 4.1

Compactly supported bounded data. For compactly supported $a \in L^{1} \cap L^{\infty}$ , choose smooth compactly supported $a_{j} \to a$ in $L^{1}$ with $∥ a_{j} ∥_{\infty} \leq ∥ a ∥_{\infty}$ , using [F5]. Their primitives converge uniformly since $\sup_{x} ∣ \int_{- \infty}^{x} (a_{j} - a) ∣ \leq ∥ a_{j} - a ∥_{1}$ . By [F4], their entropy solutions converge uniformly in time in $L^{1}$ to the solution of datum $a$ ; their normalized primitives therefore converge uniformly as well. The Hopf--Lax sup contraction in [F1] passes the identity of step 4.1 to $P (t) = Q_{t} P_{0}$ . In particular $P_{x} = u$ a.e. by [F5]. This proves (ii), including the normalization $- t f (0)$ .

6.1F1F4F5step 1.1step 5.1

A bounded Lipschitz primitive with nonintegrable derivative. Let $U_{0}$ be as in (i), and set $U_{0}^{R} (x) = U_{0} (\max {- R, \min {x, R}})$ . It has the same sup and Lipschitz bounds, and derivative $u_{0}^{R} = u_{0} 1_{(- R, R)}$ a.e. The difference between $U_{0}^{R}$ and the normalized primitive of $u_{0}^{R}$ is its constant value $U_{0} (- R)$ ; adding this constant commutes with Hopf--Lax. Thus step 5.1 shows that $(Q_{t} U_{0}^{R})_{x}$ is an entropy solution with datum $u_{0}^{R}$ . Step 1.1 places every minimiser for both $U_{0}$ and $U_{0}^{R}$ within $C t$ of $x$ . Consequently $Q_{t} U_{0}^{R} (x) = Q_{t} U_{0} (x)$ whenever $∣ x ∣ + C t < R$ , since all those competitors see identical data. Every compact positive-time cylinder is contained in such a region for large $R$ , so the a.e. derivative $v = (Q_{t} U_{0})_{x}$ is bounded by $M$ and obeys the weak equation and every entropy inequality locally, hence globally. For a compact spatial set, fix $R$ large enough that this equality holds throughout $0 \leq t \leq T$ on that set; the strong local trace of the compact-data solution supplies the trace of $v$ equal to $u_{0}$ . This proves (i), with entropy uniqueness from [F4].

7.1step 5.1step 6.1∎

Conclusion. Step 6.1 proves the derivative correspondence for the entire bounded Lipschitz primitive class, including nonintegrable derivatives, while step 5.1 proves the normalized primitive correspondence for compactly supported integrable data. In that latter class, a.e. differentiation returns $u$ , and integration from $- \infty$ with the time shift $- t f (0)$ returns the prescribed viscosity potential. All arguments hold on an arbitrary finite horizon; [F1] gives viscosity uniqueness on each such slab, while [F4] gives compatibility of the entropy solutions on overlapping horizons. These are the global solutions and inverse correspondences asserted.

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Mass conservation for compactly supported entropy solutions

Statement

Facts & Assumptions

Given: Countable and Dependent Choice, $n \geq 1$ , a $C^{1}$ flux $f$ with $f (0) = 0$ , a compactly supported datum $u_{0} \in L^{1} \cap L^{\infty}$ with $M = ∥ u_{0} ∥_{\infty}$ , the entropy solution $u$ of Existence of bounded Kruzhkov entropy solutions with its representative in $C^{0} ([0, T]; L_{l o c}^{1})$ , and a centre $x_{0} \in R^{n}$ , $R > 0$ with $u_{0} = 0$ almost everywhere outside $B (x_{0}, R)$ .

[F1]

Existence and regularity: $u$ is a bounded Kruzhkov entropy solution with $∣ u ∣ \leq M$ almost everywhere, $∥ u (t, \cdot) ∥_{\infty} \leq M$ for every $t$ , strong local $L^{1}$ trace $u_{0}$ , and an $L_{l o c}^{1}$ -continuous representative (Existence of bounded Kruzhkov entropy solutions, Kruzhkov entropy solutions).

[F2]

Finite propagation: with $L = \sup_{∣ s ∣ \leq M} ∣ f^{'} (s) ∣$ , the solution $u$ and the zero solution (which is a Kruzhkov entropy solution with datum $0$ ) agree outside the cone: $u (t, \cdot) = 0$ almost everywhere outside $B (x_{0}, R + L t)$ for almost every $t \geq 0$ ; the conclusion is an almost-everywhere statement at the level of the $L^{1}$ classes (Finite propagation for scalar conservation laws, Open ball, closed ball and sphere in a metric space).

[F3]

Weak formulation: $\int_{Π_{T}} (u φ_{t} + f (u) \cdot \nabla φ) = 0$ for every $φ \in C_{c}^{\infty} (Π_{T})$ (Distributional weak solutions of the Cauchy problem, Kruzhkov entropy solutions).

[F4]

There is a smooth compactly supported $β$ with $β = 1$ on a neighbourhood of $\overline{B} (x_{0}, R + L T)$ (Explicit compactly supported smooth cutoffs), and $L^{1}$ functions are equivalence classes, so pointwise statements on full-measure sets determine the class (The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F1F2

The cone support holds at every time for the chosen representative. By [F2] there is a full-measure set $E \subseteq (0, \infty)$ with $u (t) = 0$ almost everywhere outside $B (x_{0}, R + L t)$ for $t \in E$ . Fix $T > 0$ , $t_{0} \in [0, T]$ and a compact set $K \subseteq {x : ∣ x - x_{0} ∣ > R + L t_{0}}$ . The positive distance of $K$ from $\overline{B} (x_{0}, R + L t_{0})$ lets us choose $t_{j} \in E \cap (0, T)$ with $t_{j} \to t_{0}$ and $K \subseteq {x : ∣ x - x_{0} ∣ > R + L t_{j}}$ for all $j$ ; then $u (t_{j}) = 0$ in $L^{1} (K)$ , and the $L_{l o c}^{1}$ -continuity of the representative [F1] gives $u (t_{0}) = 0$ in $L^{1} (K)$ . A countable exhaustion of the strict exterior of $B (x_{0}, R + L t_{0})$ by compact sets gives $u (t_{0}) = 0$ almost everywhere there; hence for every $t_{0} \in [0, T]$ , $u (t_{0})$ is supported in $\overline{B} (x_{0}, R + L t_{0}) \subseteq \overline{B} (x_{0}, R + L T)$ .

2.1F3F4step 1.1

Truncated mass balance. By step 1.1, for every $t \in [0, T]$ the function $u (t, \cdot)$ vanishes almost everywhere outside the fixed ball $B (x_{0}, R + L T)$ ; in particular $u \in L^{1}$ at every time with $∥ u (t) ∥_{1} \leq M v o l (B (x_{0}, R + L T))$ , and the $L_{l o c}^{1}$ continuity of [F1] is continuity in $L^{1} (R^{n})$ . Choose $β$ as in [F4] and, for $ψ \in C_{c}^{\infty} ((0, T))$ , test [F3] with $φ (t, x) = ψ (t) β (x)$ : since $u (t, \cdot)$ is supported where $β = 1$ and $f (0) = 0$ implies $f (u) = 0$ wherever $u = 0$ , the flux term vanishes and only flat boundary terms contribute. The divergence theorem in the form of the weak identity then gives $\int_{0}^{T} ψ^{'} (t) m (t) d t = 0$ for the function $m (t) = \int_{R^{n}} u (t, x) d x$ , that is, $m^{'} = 0$ in the sense of distributions on $(0, T)$ .

3.1F1F4step 1.1step 2.1∎

Conclusion. Since $u$ is continuous in $L^{1}$ on $[0, T]$ and the support lies in the fixed ball, $m$ is continuous on $[0, T]$ ; its distributional derivative vanishes on $(0, T)$ by step 2.1 and $m (0) = \int u_{0}$ by the strong $L^{1}$ trace. To see constancy directly, convolve $m$ locally in time with a smooth unit-mass bump: its derivative is zero by $m^{'} = 0$ tested against translated kernels, so FTC makes each convolution constant on every interior compact interval. Uniform continuity of $m$ on such intervals makes the convolutions converge uniformly to $m$ , hence $m$ is constant on $(0, T)$ and by continuity at its endpoints. Therefore $m (t) = \int u_{0}$ for every $t \in [0, T]$ ; in particular the equality holds for almost every $t$ and the chosen representative makes $t \mapsto \int u (t, x) d x$ constant on every $[0, T]$ . As $T > 0$ was arbitrary, the claims follow on $[0, \infty)$ .

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An additive constant in an entropy flux does not change the entropy inequality

Statement

Let $n \geq 1$ , $f \in C^{1} (R; R^{n})$ , let $(η, q)$ be an entropy pair as in Convex entropy--entropy flux pairs, and let $q^{C} = q + C$ with a constant vector $C \in R^{n}$ . Then for every bounded measurable $u ⁣ : Π_{T} \to R$ the distributional inequalities $η (u)_{t} + {div}_{x} q (u) \leq 0 and η (u)_{t} + {div}_{x} q^{C} (u) \leq 0$ are equivalent in $D^{'} (Π_{T})$ ; the two divergences differ by the zero distribution, because the divergence of a constant vector field vanishes.

Facts & Assumptions

Given: $n \geq 1$ , an entropy pair $(η, q)$ , a constant vector $C$ , a bounded measurable $u ⁣ : Π_{T} \to R$ , and a test function $φ \in C_{c}^{\infty} (Π_{T})$ .

[F1]

The distributional divergence is defined by duality, $⟨ {div}_{x} w, φ ⟩ = - \int w \cdot \nabla φ$ , and a distribution is determined by its pairings with test functions; the test-function space is $D (Π_{T}) = C_{c}^{\infty} (Π_{T})$ (Distribution, Distributional derivative, Test function space d of an open set).

[F2]

Entropy pairs and entropy inequalities, including the dependence on the normalisation of the entropy flux, are as in Convex entropy--entropy flux pairs and Kruzhkov entropy solutions; since $q$ is locally Lipschitz and $u$ is bounded, $q (u)$ and $q^{C} (u)$ are locally integrable and their divergences are defined by [F1].

Proof

technique · direct

1.1F1given

For a test function $φ$ , [F1] and the definition of $q^{C}$ give $⟨ {div}_{x} q^{C} (u), φ ⟩ = - \int q^{C} (u) \cdot \nabla φ = - \int q (u) \cdot \nabla φ - \sum_{i = 1}^{n} C_{i} \int \partial_{x_{i}} φ$ .

1.2given

Each $\int_{Π_{T}} \partial_{x_{i}} φ d x d t = 0$ : the inner spatial integral vanishes because $φ$ is compactly supported in $x$ , so the function $x_{i} \mapsto φ (t, x_{i}, x^{'})$ is smooth compactly supported and the fundamental theorem of calculus applies, and the remaining integral over the other variables is finite as $φ$ has compact support.

2.1step 1.1step 1.2F1

By steps 1.1 and 1.2, $⟨ {div}_{x} q^{C} (u), φ ⟩ = - \int q (u) \cdot \nabla φ = ⟨ {div}_{x} q (u), φ ⟩$ for every test function, hence ${div}_{x} q^{C} (u) = {div}_{x} q (u)$ in $D^{'} (Π_{T})$ .

3.1step 2.1F2∎

Adding the common distribution $\partial_{t} η (u)$ to both sides of step 2.1, the two inequalities $\partial_{t} η (u) + {div}_{x} q (u) \leq 0$ and $\partial_{t} η (u) + {div}_{x} q^{C} (u) \leq 0$ are literally the same distributional inequality, so they are equivalent; in particular the entropy condition does not depend on the additive normalisation of the entropy flux.

Remarks

Consequently the entropy inequality depends only on the pair $(η, q)$ up to the normalisation of $q$ , and statements such as The convex entropy condition for a single shock is the chord condition are independent of the chosen constant.

CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The $L^{\infty}$ maximum bound for entropy solutions

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , $T > 0$ , let $f$ be locally Lipschitz and $C^{1}$ , and let $u$ be a bounded Kruzhkov entropy solution with initial datum $u_{0} \in L^{\infty}$ . Then, with essential extrema taken with respect to Lebesgue measure, ${ess inf}_{R^{n}} u_{0} \leq u (t, x) \leq {ess sup}_{R^{n}} u_{0} for a.e. (t, x) \in Π_{T};$ in particular $∥ u (t, \cdot) ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ for almost every $t$ . For a representative continuous in local $L^{1}$ , the same bound holds at every $t$ : local $L^{1}$ convergence from times in the full-measure set preserves the range bound (The essential supremum of a measurable function with respect to a measure, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $T > 0$ , a locally Lipschitz $C^{1}$ flux $f$ , a bounded Kruzhkov entropy solution $u$ on $Π_{T}$ with datum $u_{0} \in L^{\infty} (R^{n})$ , and the essential bounds $m_{0} = ess inf u_{0}$ , $M_{0} = ess sup u_{0}$ , both finite.

[F1]

Constant functions on $Π_{T}$ are Kruzhkov entropy solutions with their own constant value as initial datum, for every flux: for $u \equiv c$ the weak equation is the equality $\partial_{t} c + {div}_{x} f (c) = 0$ , and for every $k$ the functions $η_{k} (c) = ∣ c - k ∣$ and $q_{k} (c) = sgn (c - k) (f (c) - f (k))$ are constant in $(t, x)$ , so $\partial_{t} η_{k} (c) + {div}_{x} q_{k} (c) = 0 \leq 0$ in distributions; the strong local $L^{1}$ trace of the constant $c$ is the constant $c$ , with $\int_{K} ∣ c - c ∣ d x = 0$ for every compact $K$ (Kruzhkov entropy solutions).

[F2]

Order preservation: if two bounded Kruzhkov entropy solutions $v, w$ on $Π_{T}$ have $∣ v ∣, ∣ w ∣ \leq M$ and $v_{0} \leq w_{0}$ almost everywhere, then $v \leq w$ almost everywhere on $Π_{T}$ (Uniqueness, comparison and order preservation of entropy solutions).

[F3]

The cited essential-supremum definition defines $∥ u_{0} ∥_{\infty}$ using bounds on $∣ u_{0} ∣$ (The essential supremum of a measurable function with respect to a measure). Here define the signed extrema explicitly by $M_{0} = \inf {b \in R : u_{0} \leq b a.e.}$ and $m_{0} = \sup {a \in R : a \leq u_{0} a.e.}$ . Since $u_{0}$ is essentially bounded on the nonnull space $R^{n}$ , these are finite. For each integer $j \geq 1$ , the infimum property gives an essential upper bound below $M_{0} + 1 / j$ , so $u_{0} \leq M_{0} + 1 / j$ a.e.; the supremum property similarly gives $m_{0} - 1 / j \leq u_{0}$ a.e. Discarding the countable union of exceptional null sets and letting $j \to \infty$ yields $m_{0} \leq u_{0} \leq M_{0}$ a.e. Thus $∥ u_{0} ∥_{\infty} \leq \max {∣ m_{0} ∣, ∣ M_{0} ∣}$ . Conversely every essential absolute bound $B$ gives $m_{0} \geq - B$ and $M_{0} \leq B$ , so $\max {∣ m_{0} ∣, ∣ M_{0} ∣} \leq B$ ; taking its infimum proves equality. Inequalities between $L^{\infty}$ classes are a.e. (The space $L^{p} (μ)$ as the quotient by null functions). Fubini transfers null sets to spatial slices for a.e. time (Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

technique · direct

1.1F1F2F3

Comparison with the constant ceilings and floors. By [F1] the constants $c = M_{0}$ and $c^{'} = m_{0}$ are bounded Kruzhkov entropy solutions. Since $u_{0} \leq M_{0}$ almost everywhere and $m_{0} \leq u_{0}$ almost everywhere by [F3], choose a finite common bound for $u$ , $m_{0}$ , and $M_{0}$ . Then [F2] applied to the pairs $(u, M_{0})$ and $(m_{0}, u)$ gives $u \leq M_{0}$ almost everywhere and $m_{0} \leq u$ almost everywhere on $Π_{T}$ , that is, $m_{0} \leq u (t, x) \leq M_{0}$ for almost every $(t, x)$ .

2.1F3step 1.1

The almost-everywhere $L^{\infty}$ bound. Integrating the pointwise almost-everywhere bound of step 1.1 over spatial slices and using Fubini, for almost every $t \in (0, T)$ one has $m_{0} \leq u (t, x) \leq M_{0}$ for almost every $x$ , hence $∥ u (t, \cdot) ∥_{\infty} \leq \max {∣ m_{0} ∣, ∣ M_{0} ∣} = ∥ u_{0} ∥_{\infty}$ for almost every $t$ .

3.1F3step 2.1∎

Every time for a continuous representative. Suppose $u$ has a representative on $[0, T]$ continuous into $L_{l o c}^{1} (R^{n})$ : for $t_{j} \to t$ and every compact $K$ , $u (t_{j}) \to u (t)$ in $L^{1} (K)$ . Fix $t \in [0, T]$ and choose $t_{j} \to t$ with $t_{j}$ in the full-measure set of step 2.1. For each ball $B_{R}$ , the bound $∥ (u (t) - M_{0})_{+} ∥_{L^{1} (B_{R})} + ∥ (m_{0} - u (t))_{+} ∥_{L^{1} (B_{R})} \leq 2 ∥ u (t) - u (t_{j}) ∥_{L^{1} (B_{R})} \to 0$ preserves the range directly; exhausting $R^{n}$ by countably many balls, the bound holds for almost every $x \in R^{n}$ at this time $t$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The entropy solution semigroup on $L^{1} \cap L^{\infty}$

Statement

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the heat-kernel, $L^{1}$ completeness and vanishing-viscosity extraction interfaces used below. Let $n \geq 1$ and let $f ⁣ : R \to R^{n}$ be locally Lipschitz and $C^{1}$ . For $u_{0} \in L^{1} (R^{n}) \cap L^{\infty} (R^{n})$ and $t \geq 0$ let $S_{t} u_{0}$ be the value at time $t$ of the unique Kruzhkov entropy solution with datum $u_{0}$ (Existence of bounded Kruzhkov entropy solutions, Uniqueness, comparison and order preservation of entropy solutions), with $S_{0} u_{0} = u_{0}$ . Then: (i) $S_{t + s} = S_{t} \circ S_{s}$ for all $s, t \geq 0$ (semigroup law); (ii) each $S_{t}$ is order-preserving and an $L^{1}$ contraction, $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ ; (iii) $∥ S_{t} u_{0} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ . If in addition $f (0) = 0$ and $f$ is globally Lipschitz on $R$ , then each $S_{t}$ extends uniquely to a map on $L^{1} (R^{n})$ that is order-preserving, an $L^{1}$ contraction and satisfies the same semigroup law; the extension agrees with the classical flow on $L^{1} \cap L^{\infty}$ (Kruzhkov entropy solutions, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable and Dependent Choice, $n \geq 1$ , a locally Lipschitz $C^{1}$ flux $f$ , and data $u_{0}, v_{0} \in L^{1} \cap L^{\infty}$ with the associated unique bounded Kruzhkov entropy solutions $S_{t} u_{0}$ , $S_{t} v_{0}$ on any finite time horizon.

[F1]

Existence and uniqueness: for every datum in $L^{1} \cap L^{\infty}$ there is a bounded Kruzhkov entropy solution, unique in the bounded Kruzhkov class, with a representative continuous in $L_{l o c}^{1}$ on $[0, T]$ and attaining the datum in the strong local $L^{1}$ sense (Existence of bounded Kruzhkov entropy solutions, Uniqueness, comparison and order preservation of entropy solutions, Kruzhkov entropy solutions).

[F2]

Comparison and contraction: if $u_{0} \leq v_{0}$ almost everywhere then $S_{t} u_{0} \leq S_{t} v_{0}$ almost everywhere, and $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ for every $t$ (Uniqueness, comparison and order preservation of entropy solutions, Global $L^{1}$ contraction from the local estimate).

[F3]

$L^{\infty}$ bound: $∥ S_{t} u_{0} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ for every $t$ ; in particular the range of each solution is contained in a bounded interval on which $f$ is Lipschitz (The $L^{\infty}$ maximum bound for entropy solutions).

[F4]

Truncation and dominated convergence: for $u_{0} \in L^{1}$ , the truncations $u_{0}^{m} = (- m) \lor (u_{0} \land m)$ lie in $L^{1} \cap L^{\infty}$ and converge to $u_{0}$ in $L^{1}$ ; limits of sequences of equivalence classes are taken in $L^{1}$ and are independent of the pointwise representatives (Dominated convergence, Monotone convergence for the integral, The space $L^{p} (μ)$ as the quotient by null functions). The Cauchy limits exist by Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ .

[F5]

If an initial datum is supported in $B (0, R)$ , finite propagation gives support of its entropy solution in $B (0, R + L t)$ for almost every $t$ , where $L$ is a Lipschitz constant of $f$ on the common range (Finite propagation for scalar conservation laws, Open ball, closed ball and sphere in a metric space). The representative is continuous in $L_{l o c}^{1}$ by [F1].

Proof

technique · direct

1.1F1F2F3

Semigroup law. Fix $s, t \geq 0$ and a horizon $T > s + t$ . The solution $S_{τ} u_{0}$ is in $L^{1}$ for every $τ \in [0, T]$ : compare it with the zero solution in [F2] to get $∥ S_{τ} u_{0} ∥_{1} \leq ∥ u_{0} ∥_{1}$ ; its $L^{\infty}$ bound follows from [F3]. Thus $S_{s} u_{0} \in L^{1} \cap L^{\infty}$ and [F1] supplies the entropy solution $z (τ) = S_{τ} (S_{s} u_{0})$ . Define $w (τ, x) = S_{s + τ} u_{0} (x)$ on $Π_{T - s}$ : its entropy inequalities are those of the original solution with time shifted, and its strong local $L^{1}$ trace at $τ = 0$ is $S_{s} u_{0}$ by the representative's continuity in $L_{l o c}^{1}$ . Both $w$ and $z$ are bounded entropy solutions with this same datum, so uniqueness [F1] gives $w = z$ almost everywhere on $Π_{T - s}$ . Their time-continuous representatives then agree at every time in $L_{l o c}^{1}$ , so evaluating at $τ = t < T - s$ gives $S_{s + t} u_{0} = S_{t} (S_{s} u_{0})$ ; as $T$ is arbitrary, this holds for all $s, t \geq 0$ .

1.2F2F3

Order, contraction and the maximum bound. Let $u_{0} \leq v_{0}$ almost everywhere in $L^{1} \cap L^{\infty}$ ; by [F2] and [F3], $S_{t} u_{0} \leq S_{t} v_{0}$ almost everywhere, $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ , and $∥ S_{t} u_{0} ∥_{\infty} \leq ∥ u_{0} ∥_{\infty}$ for every $t \geq 0$ . This proves (ii) and (iii).

2.1F2F4step 1.2

Extension to $L^{1}$ : construction. Assume $f (0) = 0$ and $f$ globally Lipschitz, and let $u_{0} \in L^{1}$ . Put $u_{0}^{m} = (- m) \lor (u_{0} \land m)$ as in [F4]. For $m, ℓ \geq 1$ and every $t \geq 0$ , step 1.2 gives $∥ S_{t} u_{0}^{m} - S_{t} u_{0}^{ℓ} ∥_{1} \leq ∥ u_{0}^{m} - u_{0}^{ℓ} ∥_{1}$ , so $(S_{t} u_{0}^{m})_{m}$ is Cauchy in $L^{1}$ , uniformly in $t$ ; define $S_{t} u_{0} = \lim_{m} S_{t} u_{0}^{m}$ in $L^{1}$ . The definition is independent of the approximating sequence: if $w_{m} \in L^{1} \cap L^{\infty}$ with $w_{m} \to u_{0}$ in $L^{1}$ , then $∥ S_{t} w_{m} - S_{t} u_{0}^{m} ∥_{1} \leq ∥ w_{m} - u_{0}^{m} ∥_{1} \to 0$ , so both sequences have the same limit.

3.1F1F2F4F5step 2.1

Time continuity of the $L^{1}$ flow. First fix $w_{0} \in L^{1} \cap L^{\infty}$ and $T > 0$ , and choose $R > 0$ with $w_{0}^{R} : = w_{0} 1_{B (0, R)}$ . Let $M = ∥ w_{0} ∥_{\infty}$ and let $L$ be a Lipschitz constant of $f$ on $[- M, M]$ . By [F5], for almost every $t \in (0, T)$ the orbit $S_{t} w_{0}^{R}$ is supported in $B (0, R + L t)$ . Fix $t_{0} \in [0, T]$ and a compact set $K \subseteq {x : ∣ x ∣ > R + L t_{0}}$ . Its positive distance from $\overline{B} (0, R + L t_{0})$ lets us choose times $t_{j}$ from that full-measure set tending to $t_{0}$ with $K \subseteq {x : ∣ x ∣ > R + L t_{j}}$ . Then $S_{t_{j}} w_{0}^{R} = 0$ in $L^{1} (K)$ , and the $L_{l o c}^{1}$ continuity [F1] gives $S_{t_{0}} w_{0}^{R} = 0$ in $L^{1} (K)$ . A countable exhaustion of the strict exterior by compact sets shows that every slice is supported in $\overline{B} (0, R + L t_{0})$ ; hence all slices on $[0, T]$ are supported in the fixed ball $\overline{B} (0, R + L T)$ . Local $L^{1}$ continuity is therefore global $L^{1}$ continuity for this truncated orbit. By [F2], $\sup_{t \in [0, T]} ∥ S_{t} w_{0} - S_{t} w_{0}^{R} ∥_{1} \leq ∥ w_{0} - w_{0}^{R} ∥_{1}$ , which tends to $0$ as $R \to \infty$ . Thus the $L^{1}$ -continuous truncated orbits converge uniformly on $[0, T]$ to $t \mapsto S_{t} w_{0}$ , proving continuity for every datum in $L^{1} \cap L^{\infty}$ . For $u_{0} \in L^{1}$ in step 2.1, the extension orbit is the uniform limit of the continuous orbits $t \mapsto S_{t} u_{0}^{m}$ , since $\sup_{t \geq 0} ∥ S_{t} u_{0} - S_{t} u_{0}^{m} ∥_{1} \leq ∥ u_{0} - u_{0}^{m} ∥_{1} \to 0$ ; hence the extension is continuous as well.

3.2F2F4step 1.1step 2.1

Extension: properties. The extended maps preserve order: if $u_{0} \leq v_{0}$ in $L^{1}$ , then the truncated sequences satisfy $u_{0}^{m} \leq v_{0}^{m}$ and hence $S_{t} u_{0}^{m} \leq S_{t} v_{0}^{m}$ almost everywhere; passing to the $L^{1}$ limit gives $S_{t} u_{0} \leq S_{t} v_{0}$ almost everywhere. They are contractions: $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq {lim inf}_{m} ∥ S_{t} u_{0}^{m} - S_{t} v_{0}^{m} ∥_{1} \leq {lim inf}_{m} ∥ u_{0}^{m} - v_{0}^{m} ∥_{1} = ∥ u_{0} - v_{0} ∥_{1}$ , using that truncation is a contraction in $L^{1}$ . The semigroup law passes to the limit: $S_{t + s} u_{0} = \lim_{m} S_{t + s} u_{0}^{m} = \lim_{m} S_{t} (S_{s} u_{0}^{m}) = S_{t} (S_{s} u_{0})$ , the last step by the contraction property just proved applied to $S_{s} u_{0}^{m} \to S_{s} u_{0}$ . Finally, the extension agrees with the original flow on $L^{1} \cap L^{\infty}$ , because for such $u_{0}$ the estimate of step 2.1 with $w_{m} = u_{0}$ gives $S_{t} u_{0} = \lim_{m} S_{t} u_{0}^{m}$ in the original sense as well. An order-preserving $L^{1}$ contraction agreeing on the dense subset $L^{1} \cap L^{\infty}$ is unique, so the extension is unique.

4.1step 1.1step 1.2step 2.1step 3.1step 3.2∎

Conclusion. Steps 1.1–1.2 prove (i)–(iii) for data in $L^{1} \cap L^{\infty}$ , and steps 2.1 and 3.2 construct and characterise the unique order-preserving $L^{1}$ contraction extension to $L^{1}$ when $f (0) = 0$ and $f$ is globally Lipschitz, agreeing with the classical flow on $L^{1} \cap L^{\infty}$ and satisfying the semigroup law. Step 3.1 proves strong $L^{1}$ continuity of these orbits. This completes the proof.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Entropy solution orbits are strongly continuous in $L^{1}$

Statement

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the heat-kernel, $L^{1}$ completeness and vanishing-viscosity extraction interfaces used below. Let $n \geq 1$ , let $f ⁣ : R \to R^{n}$ be $C^{1}$ , and let $u_{0} \in L^{1} (R^{n}) \cap L^{\infty} (R^{n})$ . Then the orbit $t \mapsto S_{t} u_{0}$ is continuous from $[0, \infty)$ to $L^{1} (R^{n})$ : for every $t \geq 0$ , $∥ S_{t} u_{0} - S_{s} u_{0} ∥_{1} \to 0$ as $s \to t$ , and in particular $\lim_{t ↓ 0} ∥ S_{t} u_{0} - u_{0} ∥_{L^{1} (R^{n})} = 0.$ If additionally $f (0) = 0$ and $f$ is globally Lipschitz, its extension is a strongly continuous semigroup of $L^{1}$ contractions on all of $L^{1}$ (The entropy solution semigroup on $L^{1} \cap L^{\infty}$ , Kruzhkov entropy solutions, The space $L^{p} (μ)$ as the quotient by null functions).

Facts & Assumptions

Given: Countable and Dependent Choice, $n \geq 1$ , a $C^{1}$ flux, with the extra global Lipschitz and $f (0) = 0$ hypotheses only for the extension, a datum $u_{0} \in L^{1} \cap L^{\infty}$ with $M = ∥ u_{0} ∥_{\infty}$ , and the entropy solution semigroup $S_{t}$ of The entropy solution semigroup on $L^{1} \cap L^{\infty}$ .

[F1]

Semigroup and contraction: $S_{t + s} = S_{t} S_{s}$ , each $S_{t}$ is order-preserving, and $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ for all $t \geq 0$ and $u_{0}, v_{0} \in L^{1} \cap L^{\infty}$ ; when $f (0) = 0$ and $f$ is globally Lipschitz the maps extend to order-preserving $L^{1}$ contractions on all of $L^{1}$ , agreeing with the flow on $L^{1} \cap L^{\infty}$ (The entropy solution semigroup on $L^{1} \cap L^{\infty}$ ).

[F2]

Every any-time contraction, for data in $L^{1} \cap L^{\infty}$ , holds for every $t$ because the solutions have $L_{l o c}^{1}$ -continuous representatives: $∥ S_{t} u_{0} - S_{t} v_{0} ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ (Global $L^{1}$ contraction from the local estimate, Existence of bounded Kruzhkov entropy solutions).

[F3]

Finite propagation: if $u_{0}$ vanishes almost everywhere outside $B (0, R)$ , the entropy solution vanishes almost everywhere outside $B (0, R + L t)$ for almost every $t$ , where $L$ is a Lipschitz constant of the flux on the common range; with the $L_{l o c}^{1}$ -continuous representative this support statement upgrades to every $t$ (Finite propagation for scalar conservation laws, Open ball, closed ball and sphere in a metric space).

[F4]

The strong local $L^{1}$ trace: for every compact $K$ , ${ess sup}_{0 < t < δ} \int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x \to 0$ as $δ ↓ 0$ ; and monotone convergence controls the tails of an $L^{1}$ function over increasing balls (Kruzhkov entropy solutions, Monotone convergence for the integral, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · direct

1.1F2F3F4

Continuity at time zero. Fix $R > 0$ , put $u_{0}^{R} = u_{0} 1_{B (0, R)}$ and $v^{R} (t) = S_{t} u_{0}^{R}$ ; the datum $u_{0}^{R}$ is compactly supported, so by [F3] the solution $v^{R}$ is supported in $B (0, R + L t)$ for every $t$ , where $L$ is a Lipschitz constant of $f$ on the common range $[- ∥ u_{0} ∥_{\infty}, ∥ u_{0} ∥_{\infty}]$ . By [F2], $∥ S_{t} u_{0} - v^{R} (t) ∥_{1} \leq ∥ u_{0} - u_{0}^{R} ∥_{1} = ∥ u_{0} ∥_{L^{1} (B (0, R)^{c})}$ for every $t$ . Hence for $0 < t < 1 / L$ (any $t > 0$ when $L = 0$ ) the exterior $B (0, R + 1)^{c}$ is contained in $B (0, R + L t)^{c}$ and $∥ S_{t} u_{0} - u_{0} ∥_{1} \leq ∥ S_{t} u_{0} - u_{0} ∥_{L^{1} (B (0, R + 1))} + 2 ∥ u_{0} ∥_{L^{1} (B (0, R)^{c})},$ because on $B (0, R + 1)^{c}$ both $S_{t} u_{0}$ and $u_{0}$ have $L^{1}$ norms bounded by $∥ u_{0} ∥_{L^{1} (B (0, R)^{c})}$ (for $S_{t} u_{0}$ combine the contraction bound with $v^{R} (t) = 0$ there). The first term tends to $0$ as $t ↓ 0$ by the local $L^{1}$ continuity of the chosen representative [F2] and its trace [F4], and the tail term tends to $0$ as $R \to \infty$ by monotone convergence. Therefore $\lim_{t ↓ 0} ∥ S_{t} u_{0} - u_{0} ∥_{1} = 0$ .

2.1F1F2step 1.1

Continuity at every time. Let $s, t \geq 0$ with $t > s$ . By the semigroup law and the contraction estimate of [F1], $∥ S_{t} u_{0} - S_{s} u_{0} ∥_{1} = ∥ S_{s} (S_{t - s} u_{0}) - S_{s} u_{0} ∥_{1} \leq ∥ S_{t - s} u_{0} - u_{0} ∥_{1}$ , and the right side tends to $0$ as $t ↓ s$ by step 1.1 applied to the fixed datum $u_{0}$ . The case $s < t$ is symmetric, so the orbit is continuous at every $t \geq 0$ .

3.1F1step 2.1∎

Strong continuity on the closure. If $f (0) = 0$ and $f$ is globally Lipschitz, the extension of [F1] is defined on all of $L^{1}$ , and $L^{1} \cap L^{\infty}$ is dense in $L^{1}$ (the closure appearing in the statement). For $u_{0} \in L^{1}$ and $u_{0}^{m} = (- m) \lor (u_{0} \land m) \in L^{1} \cap L^{\infty}$ with $u_{0}^{m} \to u_{0}$ in $L^{1}$ , the contraction property gives $∥ S_{t} u_{0} - S_{t} u_{0}^{m} ∥_{1} \leq ∥ u_{0} - u_{0}^{m} ∥_{1}$ for every $t$ , so $∥ S_{t} u_{0} - S_{s} u_{0} ∥_{1} \leq 2 ∥ u_{0} - u_{0}^{m} ∥_{1} + ∥ S_{t} u_{0}^{m} - S_{s} u_{0}^{m} ∥_{1} \to 0$ by first making the two approximation errors small with a fixed large $m$ and then taking $s \to t$ , by step 2.1 applied to each $u_{0}^{m}$ . Hence the extended semigroup is strongly continuous on all of $L^{1}$ , which is the closure of $L^{1} \cap L^{\infty}$ .

5 · Examples, counterexamples and false statements

None yet.

Scalar Conservation Laws and Entropy Solutions

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Scalar conservation laws, fluxes and Cauchy data

Definition

Distributional weak solutions of the Cauchy problem

Definition

Classical solutions are distributional weak solutions, and conversely

Statement

Facts & Assumptions

Proof

Characteristics and the Riccati equation for the spatial derivative

Statement

Facts & Assumptions

Proof

Piecewise smooth shocks and one-sided traces

Definition

The Rankine--Hugoniot jump condition in space--time normal form

Statement

Facts & Assumptions

Proof

Distributional weak solutions of the Cauchy problem are not unique

Statement

Facts & Assumptions

Proof

Convex entropy--entropy flux pairs

Definition

The viscous entropy dissipation identity

Statement

Facts & Assumptions

Proof

Kruzhkov entropy solutions

Definition

The viscous scalar Cauchy problem with smooth data has a global classical solution

Statement

Facts & Assumptions

Proof

Uniform L-infinity, mass and energy bounds for the viscous approximations

Statement

Facts & Assumptions

Proof

Viscous solutions contract spatial translates in L-one

Statement

Facts & Assumptions

Proof

The Kruzhkov doubling inequality for two entropy solutions

Statement

Facts & Assumptions

Proof

Local L1L1 contraction for two entropy solutions

Statement

Facts & Assumptions

Proof

Remarks

Uniqueness, comparison and order preservation of entropy solutions

Statement

Facts & Assumptions

Proof

Finite propagation for scalar conservation laws

Statement

Facts & Assumptions

Proof

Vanishing-viscosity families are locally precompact in L1L1

Statement

Facts & Assumptions

Proof

Global L1L1 contraction from the local estimate

Statement

Facts & Assumptions

Proof

Existence of bounded Kruzhkov entropy solutions

Statement

Facts & Assumptions

Proof

The convex entropy condition for a single shock is the chord condition

Statement

Facts & Assumptions

Proof

Local $L^{1}$ contraction for two entropy solutions

Vanishing-viscosity families are locally precompact in $L^{1}$

Global $L^{1}$ contraction from the local estimate

The $L^{\infty}$ maximum bound for entropy solutions

The entropy solution semigroup on $L^{1} \cap L^{\infty}$

Entropy solution orbits are strongly continuous in $L^{1}$