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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.

Depends on

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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)