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On a Hamiltonian regularization of scalar conservation laws

Billel Guelmame

Abstract

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $\ell > 0$ and conserves an $H^1$ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $\ell$ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization. This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.

On a Hamiltonian regularization of scalar conservation laws

Abstract

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by and conserves an energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization. This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.
Paper Structure (10 sections, 24 theorems, 174 equations)

This paper contains 10 sections, 24 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Variational formulations
  3. Main results
  4. The approximated equation
  5. Precompactness of the approximated equation
  6. The limiting case $\ell \to 0$
  7. Uniform estimates for small $\ell$
  8. Precompactness
  9. The limiting case $\ell \to \infty$
  10. Some classical lemmas

Key Result

Theorem 3.2

Let $f \in C^4$ be a uniformly convex flux ($f"(u) \geqslant c >0$), $u_0 \in H^1(\mathds{R})$ and $\ell>0$, then there exists a global weak dissipative solution $u^\ell \in L^\infty ([0,\infty ), H^1(\mathds{R})) \cap C([0,\infty ) \times \mathds{R})$ of rB in the sense of Definition WSDef satisfyi Moreover, if $f"(u) \leqslant C$, $u_0' \in L^1(\mathds{R})$ and $u_0'(x) \leqslant M < \infty$ the

Theorems & Definitions (30)

  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Definition 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 20 more