Global weak solutions of a Hamiltonian regularised Burgers equation

TL;DR

This paper introduces a new regularisation of the inviscid Burgers equation that admits weak solutions with shocks and cusps, analyzing their existence, properties, and limits as the regularisation parameter varies.

Contribution

It proposes a novel Hamiltonian regularisation of Burgers, establishing existence of weak solutions, and explores their behavior and limits, connecting to Burgers and Hunter-Saxton equations.

Findings

Existence of weak solutions with shocks and cusps

Dissipative solutions satisfy Oleinik-type inequalities

Limits recover Burgers or Hunter-Saxton equations as regularisation scale varies

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.