Convergence rate for a regularized scalar conservation law

This work revisits a recent finding by the first author concerning the local convergence of a regularized scalar conservation law. We significantly improve the original statement by establishing a global convergence result within the Lebesgue spaces \(L^\infty _{\textrm{loc}}(\mathbb {R}^+;L^p(\mathbb {R}))\), for any \(p \in [1,\infty )\), as the regularization parameter \(\ell \) approaches zero. Notably, we demonstrate that this stability result is accompanied by a quantifiable rate of convergence. A key insight in our proof lies in the observation that the fluctuations of the solutions remain under control in low regularity spaces, allowing for a potential quantification of their behavior in the limit as \(\ell \rightarrow 0\). This is achieved through a careful asymptotic analysis of the perturbative terms in the regularized equation, which, in our view, constitutes a pivotal contribution to the core findings of this paper.