Regularizing effect for conservation laws with a Lipschitz convex flux

This paper studies the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, we generalize the well-known $\operatorname{BV}$ smoothing effect for $\mathrm{C}^2$ uniformly convex fluxes discovered independently by [P.D. Lax, Comm. Pure Appl. Math., 10:537–566, 1957] and [O. Oleinik, Amer. Math. Soc. Transl., 26(2):95–172, 1963], while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be discontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized $\operatorname{BV}$ space, $\operatorname{BV} \phi$.

Keywords

scalar conservation laws, entropy solution, strictly convex flux, discontinuous velocity, wave front tracking, smoothing effect, $\operatorname{BV} \phi$ spaces

2010 Mathematics Subject Classification

26A45, 35B65, 35L65, 35L67, 46E30

Full Text (PDF format)

Received 17 December 2018

Accepted 31 July 2019

Published 3 February 2020