On a Hamiltonian regularization of scalar conservation laws

In this paper, we study a regularization of a scalar conservation law (SCL), which is obtained by modifying its Lagrangian. This regularization is parameterized by $\ell$ and conserves formally an $H^1$-like energy. Proof of the existence of local smooth solutions are given in this paper. In addition, we prove the existence of global weak solutions satisfying a uniform (on $\ell$) one-sided Oleinik inequality for this regularization, and also for a generalized Hunter--Saxton equation. Moreover, when $\ell \to 0$ (resp. $\ell \to \infty$), we prove that the solutions of the regularized equation converge up to a subsequence to $u^0$ (resp.$ u^\infty$) a solution of the SCL (resp. a generalized Hunter--Saxton equation), at least before the appearance of singularities