Non existence of the BV regularizing effect for scalar conservation laws in several space dimension

TL;DR

This paper demonstrates that for certain scalar conservation laws in multiple dimensions, entropy solutions can lack BV regularity, challenging previous assumptions about regularizing effects of flux functions.

Contribution

It constructs explicit examples of non-BV entropy solutions for a broad class of C2 fluxes, showing the non-existence of BV regularizing effects in several space dimensions.

Findings

Existence of non-BV entropy solutions in multi-D for certain fluxes

Classification of C2 fluxes based on BV regularity of solutions

Extension of results to fractional Sobolev spaces for non-degenerate fluxes

Abstract

This article deals with the regularity aspects of entropy solutions to scalar conservation laws. We show that for each C2 flux in multi-D, there exists an entropy solution which does not belong to BV locally for all time. For this purpose, we construct a non-BVloc solution in 1-D for a special class of C2 fluxes whose second derivative has a zero. It covers all the C2 functions for which Lax-Oleinik's BV regularizing result is not applicable and provides a classification of one dimensional C2 fluxes based on L\infty-BVloc regularizing of entropy solution. In the later part of this article, we extend our result to fractional Sobolev spaces for a class of non-degenerate fluxes.