Oscillation properties of scalar conservation laws

TL;DR

This paper establishes new regularity and decay properties of solutions to scalar conservation laws under the genuine nonlinearity condition, including continuity outside jump sets and algebraic decay rates.

Contribution

It introduces a novel local oscillation estimate inspired by De Giorgi's methods, leading to improved regularity and decay results for scalar conservation laws.

Findings

Solutions are continuous outside the jump set

Entropy dissipation vanishes away from the jump set

Solutions decay algebraically in $L^inity$ as time increases

Abstract

We obtain several new regularity results for solutions of scalar conservation laws satisfying the genuine nonlinearity condition. We prove that the solutions are continuous outside of the jump set, which is codimension one rectifiable. We show that the entropy dissipation vanishes away from the closure of the jump set. We prove that the solution decays algebraically in $L^\infty$ as $t \to \infty$ and we compute the presumably optimal decay rate. All these results are based on a local oscillation estimate which is obtained properly adapting some ideas of De Giorgi from the context of elliptic equations.