BV regularity near the interface for nonuniform convex discontinuous flux

TL;DR

This paper establishes BV regularity near interfaces for scalar conservation laws with discontinuous flux, demonstrating a smoothing effect without requiring uniform convexity of the flux functions.

Contribution

It proves BV regularity near interfaces for solutions of scalar conservation laws with discontinuous flux without assuming flux convexity, using characteristics and explicit formulas.

Findings

BV regularity near the interface is achieved for solutions with bounded initial data.

The smoothing effect occurs without the need for uniform convexity of flux functions.

Explicit formulas and characteristics are key tools in the proof.

Abstract

In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.