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 $H^1$ 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.

arXiv commentPublished online in J. Dynam. Differential Equations in 2022

References in corpus (6)

• An integrable shallow water equation with peaked solitons
• On the Galilean invariance of some dispersive wave equations
• Non-dispersive conservative regularisation of nonlinear shallow water (and isothermal Euler) equations
• Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations
• Well-posedness and derivative blow-up for a dispersionless regularized shallow water system
• Hamiltonian regularisation of the unidimensional barotropic Euler equations

Cited by in corpus (4)

• Hamiltonian regularisation of the unidimensional barotropic Euler equations
• On a Hamiltonian regularization of scalar conservation laws
• On the blow-up scenario for some modified Serre-Green-Naghdi equations
• Parameter Estimation and Adaptive Solution of the Leray-Burgers Equation Using Physics-Informed Neural Networks