Schaeffer's regularity theorem for scalar conservation laws does not extend to systems

TL;DR

This paper demonstrates that Schaeffer's regularity theorem, which guarantees smooth solutions outside finitely many curves for scalar conservation laws with convex flux, does not extend to systems, via a counterexample.

Contribution

The paper provides the first explicit counterexample showing the failure of Schaeffer's regularity theorem for systems of conservation laws.

Findings

Counterexample disproves extension of Schaeffer's theorem to systems

Interaction estimates are crucial in the analysis

Wave front-tracking approximation reveals the irregularity

Abstract

Schaeffer's regularity theorem for scalar conservation laws can be loosely speaking formulated as follows. Assume that the flux is uniformly convex, then for a generic smooth initial datum the admissible solution is smooth outside a locally finite number of curves in the (t,x) plane. Here the term `generic' is to be interpreted in a suitable sense, related to the Baire Category Theorem. Whereas other regularity results valid for scalar conservation laws with convex fluxes have been extended to systems of conservation laws with genuinely nonlinear characteristic fields, in this work we exhibit an explicit counterexample which rules out the possibility of extending Schaeffer's Theorem. The analysis relies on careful interaction estimates and uses fine properties of the wave front-tracking approximation.