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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 $L^{1} \cap L^{\infty}$ .

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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)
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)

Entropy solution orbits are strongly continuous in L1L1