SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1908-1943, January 2021.
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.

中文翻译：

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标量守恒律在几个空间维度上对于$ C ^ 2 $通量的$ BV $正则化效应的不存在

SIAM数学分析杂志，第53卷，第2期，第1908-1943页，2021年1月。
本文讨论标量守恒定律的熵解的规则性方面。我们表明，对于多维（multi-D）中的每个$ C ^ 2 $通量，存在一个熵解，该熵解在所有时间内都不属于$ BV_ {loc}（\ mathbb {R} ^ d）$。为此，对于一类特殊的$ C ^ 2 $通量，其二阶导数为零，我们在一维（1D）中构造了一个非$ BV_ {loc} $解决方案。它涵盖了Lax-Ole \ u \ inik $ BV $正则化结果不适用的所有$ C ^ 2 $函数，并基于$ L ^ \ infty提供了一维$ C ^ 2 $通量的分类$-$ BV_ {loc} $熵解的正则化。在本文的后半部分，我们将结果扩展到一类非退化通量的分数Sobolev空间。