Global weak solutions of a Hamiltonian regularised Burgers equation

Billel Guelmame (1, 2) , Stéphane Junca (1, 2) , Didier Clamond (1) , Robert L Pego (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 COFFEE - COmplex Flows For Energy and Environment
3 CNA - Center for Nonlinear Analysis [Pittsburgh]

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.

Mots clés: Inviscid Burgers equation; regularisation; Hamiltonian; conservative and dissipative solutions; Oleinik inequality

Domaines: Equations aux dérivées partielles [math.AP]

Billel Guelmame, Stéphane Junca, Didier Clamond, Robert L Pego. Global weak solutions of a Hamiltonian regularised Burgers equation. Journal of Dynamics and Differential Equations, 2022, https://link.springer.com/article/10.1007/s10884-022-10171-0. ⟨10.1007/s10884-022-10171-0⟩. ⟨hal-02478872v2⟩