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Affine flux reduces the entropy semigroup to translation

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), as used by the translation-continuity and entropy-semigroup results below. Let $f (u) = c u + d$ , with $c, d \in R$ , and let $u_{0} \in L^{1} (R) \cap L^{\infty} (R)$ . The entropy solution is $u (t, x) = u_{0} (x - c t) .$ For every convex entropy pair $(η, q)$ , $q^{'} (u) = c η^{'} (u)$ , hence $q (u) = c η (u) + C$ ; the transport change of variables gives $\partial_{t} η (u) + \partial_{x} q (u) = 0$ in distributions, so entropy production is zero. Any jump already present in $u_{0}$ translates at speed $c$ with the same left and right states and zero production $[q] - c [η] = 0$ ; the affine evolution creates no new shocks. In particular the semigroup of The entropy solution semigroup on $L^{1} \cap L^{\infty}$ is $S_{t} u_{0} = u_{0} (\cdot - c t)$ (Scalar conservation laws, fluxes and Cauchy data, Kruzhkov entropy solutions, Distributional weak solutions of the Cauchy problem).

Facts & Assumptions

Given: an affine flux $f (u) = c u + d$ , a datum $u_{0} \in L^{1} (R) \cap L^{\infty} (R)$ , the translated profile $u (t, x) = u_{0} (x - c t)$ , and a test function $φ \in C_{c}^{\infty} (Π_{T})$ .

[F1]

Translation invariance: the maps $y \mapsto y + c t$ preserve Lebesgue measure; integrals of integrable functions are invariant under measure-preserving transformations, so $\int_{R} v (x - c t) d x = \int_{R} v (y) d y$ , and with Fubini the substitution $y = x - c t$ is legitimate in the space--time integrals below (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Measure-preserving transformations and systems, Integral invariance under measure-preserving maps, Fubini's theorem for L^1 functions on a sigma-finite product, Translation of a function on $R^{n}$ ).

[F2]

Calculus: the chain rule gives $\partial_{t} [φ (t, y + c t)] = φ_{t} (t, y + c t) + c φ_{x} (t, y + c t)$ at $y = x - c t$ , and the fundamental theorem of calculus with compact support gives $\int_{R} φ_{x} (t, x) d x = 0$ and $\int_{0}^{T} φ_{t} (t, x) d t = 0$ for compactly supported $φ$ (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F3]

Entropy pairs for an affine flux: $q^{'} = η^{'} f^{'} = c η^{'}$ , so $q = c η + C$ on the real line; a jump of the translated profile has $[q] = c [η] + [C] = c [η]$ , hence zero production $[q] - c [η] = 0$ (Convex entropy--entropy flux pairs, Kruzhkov entropy solutions).

[F4]

The semigroup: for a locally Lipschitz $C^{1}$ flux and data in $L^{1} \cap L^{\infty}$ the entropy solution is unique and the flow defines $S_{t}$ ; for $f (0) = 0$ globally Lipschitz it extends to all of $L^{1}$ (The entropy solution semigroup on $L^{1} \cap L^{\infty}$ , 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); translation is continuous in $L^{1}$ , so $∥ u_{0} (\cdot - c t) - u_{0} ∥_{1} \to 0$ as $t ↓ 0$ ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ , Translation of a function on $R^{n}$ ).

Proof

technique · direct

1.1F1F2

The translate solves the conservation law. Substituting $y = x - c t$ in the weak pairing and using [F1]: $\int_{Π_{T}} (u φ_{t} + f (u) φ_{x}) d x d t = \int_{Π_{T}} (u_{0} (y) φ_{t} (t, y + c t) + c u_{0} (y) φ_{x} (t, y + c t) + d φ_{x} (t, y + c t)) d y d t$ . The constant term vanishes, by [F2] applied to $φ_{x}$ ; the remaining terms equal $\int_{Π_{T}} u_{0} (y) \partial_{t} [φ (t, y + c t)] d y d t$ by the chain rule, which is $0$ because $φ$ has compact support in time. Hence $u$ is a distributional weak solution of $u_{t} + \partial_{x} (c u + d) = 0$ .

1.2F1F2F3

Zero entropy production. Let $(η, q)$ be any locally Lipschitz convex pair with $q^{'} = η^{'} f^{'} = c η^{'}$ , so $q = c η + C$ by [F3]. Applying the same substitution to $η (u (t, x)) = η (u_{0} (x - c t))$ and $q (u) = c η (u_{0} (x - c t)) + C$ gives $\int_{Π_{T}} (η (u) φ_{t} + q (u) φ_{x}) = \int_{Π_{T}} η (u_{0} (y)) [φ_{t} (t, y + c t) + c φ_{x} (t, y + c t)] d y d t + C \int_{Π_{T}} φ_{x} = \int_{Π_{T}} η (u_{0} (y)) \partial_{t} [φ (t, y + c t)] d y d t + 0 = 0$ , using the chain rule [F2] and the vanishing of the constant term there. Thus the entropy production vanishes in distributions; for a jump already present in $u_{0}$ this is the statement $[q] - c [η] = 0$ of [F3], so the jump keeps its states and produces no dissipation.

2.1F3F4step 1.1step 1.2∎

Initial trace and identification with the semigroup. The initial trace is immediate: $u (t, x) = u_{0} (x - c t)$ gives $∥ u (t, \cdot) - u_{0} ∥_{1} = ∥ u_{0} (\cdot - c t) - u_{0} ∥_{1} \to 0$ as $t ↓ 0$ by translation continuity in $L^{1}$ [F4]. The profile is therefore a Kruzhkov entropy solution with datum $u_{0}$ (weak equation, all entropy inequalities, strong trace), and by uniqueness in [F4] it agrees with the semigroup flow: $S_{t} u_{0} = u_{0} (\cdot - c t)$ for all $t \geq 0$ .

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Sources

S. N. Kruzhkov, “First order quasilinear equations in several independent variables,” Mat. USSR-Sbornik 10 (1970), 217–243, complete English translation (standard reference, not scraped)
Alberto Bressan, “Hyperbolic Conservation Laws: An Illustrated Tutorial,” 2009, complete lecture notes (standard reference, not scraped)