Optimal regularity for all time for entropy solutions of conservation laws in $B{V}^s$

Abstract

This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional $BV$ spaces. We show that optimality of the regularizing effect for the initial value problem from $L^{\infty }$ to fractional Sobolev space and fractional $BV$ spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover , we prove sharp smoothing in $B{V}^s$ for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

Published in NoDEA in 2020

References in corpus (4)

• Regularizing effect for conservation laws with a Lipschitz convex flux
• Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws
• Entropy solutions in $B{V}^s$ for a class of triangular systems involving a transport equation
• Structure and regularity of solutions to 1d scalar conservation laws

Cited by in corpus (1)

• Entropy solutions in $B{V}^s$ for a class of triangular systems involving a transport equation