Higher regularity for entropy solutions of conservation laws with geometrically constrained discontinuous flux

TL;DR

This paper establishes optimal geometric conditions under which entropy solutions of conservation laws with discontinuous flux attain regularity comparable to smooth flux cases, improving understanding of solution behavior.

Contribution

It introduces a geometric condition that recovers higher regularity for entropy solutions with discontinuous flux, matching the regularity of smooth flux scenarios.

Findings

Optimal geometric condition recovers BV regularity

Regularity of solutions matches smooth flux case

Fractional regularity s=1/2 achieved under new conditions

Abstract

For the Burgers equation, the entropy solution becomes instantly BV with only $L^\infty$ initial data. For conservation laws with genuinely nonlinear discontinuous flux, it is well known that the BV regularity of entropy solutions is lost. Recently, this regularity has been proved to be fractional with s = 1/2. Moreover, for less nonlinear flux the solution has still a fractional regularity 0 < s \leq 1/2. The resulting general rule is the regularity of entropy solutions for a discontinuous flux is less than for a smooth flux. In this paper, an optimal geometric condition on the discontinuous flux is used to recover the same regularity as for the smooth flux with the same kind of nonlinearity.