High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws

TL;DR

This paper investigates high frequency wave propagation in nonlinear scalar conservation laws and demonstrates that the maximal smoothing effect predicted by a longstanding conjecture cannot be surpassed.

Contribution

It introduces a new definition of nonlinear flux and proves the maximal regularity limit for solutions, confirming a conjecture about the smoothing effect.

Findings

High frequency waves propagate near constant solutions.

The maximal Sobolev regularity predicted cannot be exceeded.

A new flux definition clarifies the regularization limits.

Abstract

The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, (1994), about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a new definition of nonlinear flux is stated and compared to classical definitions. Then it is proved that the smoothness expected in Sobolev spaces cannot be exceeded.