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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).

Depends on

Used by

The L^∞ maximum bound for entropy solutions Corollary
Rankine--Hugoniot alone does not give uniqueness Counterexample
The convex-flux Riemann formula fails for a nonconvex flux Counterexample
Gradient catastrophe before shock formation Example
Nonconvex Riemann data can require a composite shock--rarefaction wave Example
Existence of bounded Kruzhkov entropy solutions Theorem
The entropy solution semigroup on L¹∩ L^∞ Theorem
The Hamilton--Jacobi correspondence in one dimension Theorem
The Riemann solver for a strictly convex flux Theorem

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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)
G. A. Chechkin and A. Yu. Goritsky (translated by B. Andreianov), “S. N. Kruzhkov’s lectures on first-order quasilinear PDEs,” in Analytical and Numerical Aspects of PDEs, de Gruyter 2009, complete lecture-notes text (standard reference, not scraped)