Schaeffer's Regularity Theorem for Scalar Conservation Laws Does Not Extend to Systems

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 show an explicit counterexample that 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.