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

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.

Cited by in corpus (4)

• Regularity estimates for scalar conservation laws in one space dimension
• Regularizing effect for conservation laws with a Lipschitz convex flux
• Optimal regularity for all time for entropy solutions of conservation laws in $B{V}^s$
• High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws