A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation

TL;DR

This paper reviews regularity results for Burgers' equation and explores their limitations in genuinely nonlinear hyperbolic systems, including an original counterexample illustrating shock pattern stability.

Contribution

It discusses the extension of Burgers' equation regularity results to systems and provides a new counterexample demonstrating failure of regularity in complex shock patterns.

Findings

Oleinik-Ambroso-De Lellis SBV estimate applies to Burgers' equation

Schaeffer's theorem indicates generic initial data lead to piecewise smooth solutions

Counterexample shows stability of shock patterns with infinitely many shocks

Abstract

The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the x-derivative of u when u is an entropy solution of the Cauchy problem for Burgers' equation with bounded initial data. Its extensions to the case of systems is then mentioned. The second regularity result of debate is Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and generic, in a Baire category sense, initial data are piecewise smooth. The failure of the same regularity for general genuinely nonlinear systems is next described. The main focus of this paper is indeed including heuristically an original counterexample where a kind of stability of a shock pattern made by infinitely many shocks shows up, referring to [Caravenna-Spinolo] for the rigorous result.