Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws

TL;DR

This paper investigates the optimal smoothing effects for one-dimensional nonlinear scalar conservation laws, introducing new definitions and fractional BV spaces to analyze entropy solutions and their regularity.

Contribution

It provides new definitions of nonlinear flux and introduces fractional BV spaces to characterize the regularity of solutions in one-dimensional scalar conservation laws.

Findings

Supercritical geometric optics produce smooth solutions bounded in the conjectured Sobolev space.

Constructed continuous solutions exactly in the appropriate Sobolev space.

Demonstrated the relevance of fractional BV spaces for analyzing solution regularity.

Abstract

Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for entropy solutions of nonlinear scalar conservations laws . In this short paper we will restrict our attention to the simpler one-dimensional case. First, supercritical geometric optics lead to sequences of $C^\infty$ solutions uniformly bounded in the Sobolev space conjectured. Second we give continuous solutions which belong exactly to the suitable Sobolev space. In order to do so we give two new definitions of nonlinear flux and we introduce fractional $BV$ spaces.