Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

Abstract

This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional BV spaces. We show that optimality of the regularizing effect for the initial value problem from $$L^\infty $$ to fractional Sobolev space and fractional BV spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in $$BV^s$$ for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.