Fractional regularity for conservation laws with discontinuous flux

Abstract

This article deals with the regularity of entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such an equation does not admit BV regularity in general, even when the initial data belongs to BV. Due to this phenomenon, fractional BVs spaces, where the exponent 0<s≤1 and BV=BV1, are required to be wider than BV. It is a long-standing open question to find the optimal regularizing effect for the discontinuous flux with L∞ initial data. The optimal regularizing effect in BVs is proven in an important case using control theory, and the fractional exponent s is at most 1/2, even when the fluxes are uniformly convex.