Nonexistence of the $BV$ Regularizing Effect for Scalar Conservation Laws in Several Space Dimensions for $C^2$ Fluxes

Source abstract

This article deals with the regularity aspects of entropy solutions to scalar conservation laws. We show that for each $C^2$ flux in multiple dimensions (multi-D), there exists an entropy solution which does not belong to $BV_{loc}(\mathbb{R}^d)$ for all time. For this purpose, we construct a non-$BV_{loc}$ solution in one dimenstion (1D) for a special class of $C^2$ fluxes whose second derivative has a zero. It covers all the $C^2$ functions for which the Lax--Ole\u\inik $BV$ regularizing result is not applicable and provides a classification of one-dimensional $C^2$ fluxes based on $L^\infty$-$BV_{loc}$ regularizing of the entropy solution. In the latter part of this article, we extend our result to fractional Sobolev spaces for a class of nondegenerate fluxes.