Regularity estimates for scalar conservation laws in one space dimension

TL;DR

This paper investigates how the nonlinearity of the flux function in scalar conservation laws influences the regularity of entropy solutions, establishing new regularity results in BV and SBV spaces under specific conditions.

Contribution

It introduces novel regularity estimates for entropy solutions based on the flux function's nonlinearity, including BV and SBV regularity results under degeneracy conditions.

Findings

Regularity of solutions expressed in BV^Φ spaces depending on flux nonlinearity.

Proved f'∘u(t) belongs to BV_loc and SBV_loc spaces under certain degeneracy conditions.

Examples demonstrating the sharpness of the regularity results and counterexamples to related properties.

Abstract

In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function $f$ has on the entropy solution. More precisely, if the set $\{w:f''(w)\ne 0\}$ is dense, the regularity of the solution can be expressed in terms of $\mathrm{BV}^\Phi$ spaces, where $\Phi$ depends on the nonlinearity of $f$. If moreover the set $\{w:f''(w)=0\}$ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that $f'\circ u(t)\in \mathrm{BV}_{\mathrm{loc}}(\mathrm{R})$ for every $t>0$ and that this can be improved to $\mathrm{SBV}_{\mathrm{loc}}(\mathbb{R})$ regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.