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

Several regularity results hold for the Cauchy problem involving one scalar conservation law having convex flux. Among these, Schaeffer's theorem guarantees that if the initial datum is smooth and is generic, in the Baire sense, the entropy admissible solution develops at most finitely many shocks, locally, and stays smooth out of them. We rule out with the present paper the possibility of extending Schaeffer's regularity result to the class of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The analysis relies on careful interaction estimates and uses fine properties of the wave-front tracking approximation.