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Journal Articles Journal of Dynamics and Differential Equations Year : 2022

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 equationand 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.
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Dates and versions

hal-02478872 , version 1 (14-02-2020)
hal-02478872 , version 2 (10-03-2022)

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Cite

Billel Guelmame, Stéphane Junca, Didier Clamond, Robert L Pego. Global weak solutions of a Hamiltonian regularised Burgers equation. Journal of Dynamics and Differential Equations, inPress, ⟨10.1007/s10884-022-10171-0⟩. ⟨hal-02478872v2⟩
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