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Contractive metrics for scalar conservation laws

F. Bolley, Y. Brenier, G. Loeper

TL;DR

The paper analyzes nondecreasing entropy solutions to 1-D scalar conservation laws and shows that their spatial derivatives, viewed as probability measures, satisfy a contraction with respect to the Wasserstein distance $W_p$ of any order, extending Kruzhkov's $L^1$ contraction to the derivative variable. The authors introduce an inverse-profile representation $X(t,w)$, solving $X_t=f'(u(t,X))$ so that $u(t,X(t,w))=w$ and $X_t(t,w)=f'(w)$, which yields $X- ilde X$ constant and hence time-invariance of $W_p(u_x, ilde u_x)$ for classical increasing solutions (and, for $p=1$, Kruzhkov-type contraction for the original profiles). They develop a time-discretization (Transport-Collapse) scheme with operators $T_h$ and $S_h$ that preserves the admissible classes, provides $W_p$-contraction for the spatial derivatives, and converges so $[S_h u^0]_x(t,.) o u_x(t,.)$ in $W_p$. The framework is extended to viscous conservation laws, establishing existence, uniqueness, and the same $W_p$ contraction for $u_x(t,.)$, with the inviscid limit obtained via vanishing viscosity and a discrete viscous scheme built from a heat kernel that preserves Wasserstein distances.

Abstract

We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L^1-contraction property shown by Kruzkov.

Contractive metrics for scalar conservation laws

TL;DR

The paper analyzes nondecreasing entropy solutions to 1-D scalar conservation laws and shows that their spatial derivatives, viewed as probability measures, satisfy a contraction with respect to the Wasserstein distance of any order, extending Kruzhkov's contraction to the derivative variable. The authors introduce an inverse-profile representation , solving so that and , which yields constant and hence time-invariance of for classical increasing solutions (and, for , Kruzhkov-type contraction for the original profiles). They develop a time-discretization (Transport-Collapse) scheme with operators and that preserves the admissible classes, provides -contraction for the spatial derivatives, and converges so in . The framework is extended to viscous conservation laws, establishing existence, uniqueness, and the same contraction for , with the inviscid limit obtained via vanishing viscosity and a discrete viscous scheme built from a heat kernel that preserves Wasserstein distances.

Abstract

We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L^1-contraction property shown by Kruzkov.

Paper Structure

This paper contains 14 sections, 19 theorems, 78 equations.

Table of Contents

  1. Introduction to the results
  2. Wasserstein distances
  3. The case of classical solutions: Theorem \ref{['theoremeclassique']} and corollary
  4. Proof of Theorem \ref{['theoremeclassique']}
  5. Proof of Corollary \ref{['corclassique']}
  6. Time discretization of the conservation law
  7. Definition and $W_p$ contraction property of the discretized solution
  8. Convergence of the scheme in the $L^{1}_{loc}(\mathbb R)$ sense
  9. Convergence of the scheme in $W_p$ distance sense
  10. The general case of entropy solutions: Theorem \ref{['theoreme']} and corollaries
  11. Proof of Theorem \ref{['theoreme']}
  12. Proof of Corollary \ref{['corsolutionUp']}
  13. Proof of Corollary \ref{['corsolutionU1']}
  14. Extension to viscous conservation laws

Key Result

Theorem 1.1

For every u^{0} \in L^{\infty}(\mathbb R), there exists a unique entropy solution u to $(cons law)$ in L^{\infty}([0,+\infty[ \times \mathbb R) \cap \mathcal{C} ([0,+\infty[, L_{loc}^{1}(\mathbb R)).

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 14 more