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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 $∥ u (t) - v (t) ∥_{1} \leq ∥ u_{0} - v_{0} ∥_{1}$ .

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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)

Global L1L1 contraction from the local estimate