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Regularizing effect for conservation laws with a Lipschitz convex flux

Billel Guelmame, Stéphane Junca, Didier Clamond

Abstract

This paper studies the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, we generalize the well-known $\mathrm{BV}$ smoothing effect for $\mathrm{C}^2$ uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik, while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be dicontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized $\mathrm{BV}$ space, $\mathrm{BV}^Φ$.

Regularizing effect for conservation laws with a Lipschitz convex flux

Abstract

This paper studies the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on . Beside convexity, no additional regularity is assumed on the flux. Thus, we generalize the well-known smoothing effect for uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik, while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be dicontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized space, .
Paper Structure (8 sections, 5 theorems, 60 equations)

This paper contains 8 sections, 5 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Notes on the Oleinik inequality for discontinuous wave speeds
  4. Riemann problem
  5. The exact solution of the Riemann problem
  6. Approximate Riemann solver
  7. Wave front tracking algorithm
  8. Compactness and regularity

Key Result

Proposition 2.1

Billel

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Proposition 2.1: see Proposition 2.3 in inverse
  • Definition 2.5
  • Remark 2.2
  • Theorem 2.1
  • Proposition 3.1
  • ...and 13 more