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Global weak solutions of a Hamiltonian regularised Burgers equation

Billel Guelmame, Stéphane Junca, Didier Clamond, Robert L. Pego

Abstract

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term.

Global weak solutions of a Hamiltonian regularised Burgers equation

Abstract

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter tends to or , limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term.
Paper Structure (16 sections, 15 theorems, 162 equations, 2 figures)

This paper contains 16 sections, 15 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Heuristic derivation of a regularised Burgers equation
  3. Existence and breakdown of smooth solutions
  4. Global weak solutions: conservative case
  5. Global weak solutions: dissipative case
  6. Traveling waves of permanent form
  7. Local analysis of weak solutions
  8. Cuspons and periodic cuspons
  9. Shock layers
  10. Composite waves
  11. Non-existing waves
  12. The limiting cases $\ell\to0$ and $\ell\to+\infty$ for dissipative solutions
  13. The limiting case $\ell \to 0$
  14. The limiting case $\ell \to +\infty$
  15. Optimality of the $\dot{H}_{loc}$ space
  16. ...and 1 more sections

Key Result

Theorem 3.1

For an initial datum $u_0 \in H^s(\mathds{R})$ with $s>3/2$, there exists a maximal time $T^*>0$ (independent of $s$) and a unique solution $u \in \mathcal{C}([0,T^*[,H^s)$ of RB_2 such that (blow-up criterium) Moreover, if $s \geqslant 3$, then

Figures (2)

  • Figure 1: Types of weakly singular stationary waves: (a) cuspon; (b) periodic cuspon; (c) shock layer; (d) composite wave
  • Figure 2: Regions where $v=-\pi$.

Theorems & Definitions (30)

  • Theorem 3.1: Yin2004byin2007cauchy
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Remark 3.6
  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • ...and 20 more