Global weak solutions of a Hamiltonian regularised Burgers equation

Inspired by a recent nondispersive conservative regularisation of the shallow water equations, a similar regularisation is proposed and studied here for the inviscid Burgers equation. The regularised equation is parametrised by a positive number $\ell$, the inviscid Burgers equation corresponding to $\ell=0$ and the Hunter--Saxton equation being formally obtained letting $\ell\to \infty$. The breakdown of local smooth solutions is demonstrated. The existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is also studied. The built {\em dissipative} solution satisfies (uniformly with respect to $\ell$) an Oleinik inequality, as do entropy solutions of the inviscid Burgers equation. The limit (up to a subsequence) of the dissipative solution when $\ell\to 0$ (respectively $\ell \to \infty$) satisfies the Burgers (resp. Hunter--Saxton) equation forced by an unknown remaining term. At least before the appearance of singularities, the limit satisfies the Burgers (resp. Hunter--Saxton) equation.