Fractional regularity for conservation laws with discontinuous flux

TL;DR

This paper investigates the fractional regularity of entropy solutions to scalar conservation laws with discontinuous flux, establishing that the optimal fractional BV regularity exponent is at most 1/2, even with convex fluxes.

Contribution

It proves the optimal fractional BV regularity exponent for solutions with discontinuous flux, advancing understanding of regularity in such conservation laws.

Findings

Fractional BV regularity is necessary for solutions with discontinuous flux.

The optimal fractional exponent s is at most 1/2.

Regularity results hold even for uniformly convex fluxes.

Abstract

This article deals with the regularity of the 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 equation does not admit BV regularity in general, even when the initial data belongs to BV. Due to this phenomenon fractional BVs spaces wider than BV are required, where the exponent 0<s\leq 1 and BV = BV1. It is a long standing open question to find the optimal regularizing effect for the discontinuous flux with L^\infty initial data. The optimal regularizing effect in BVs is proven on an important case using control theory. The fractional exponent s is at most 1/2 even when the fluxes are uniformly convex.