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

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 Ole ̆ınik-Ambroso-De Lellis SBV estimate: it provides bounds on ∂xu when u is an entropy solution of the Cauchy problem for Burgers’ equation with L∞-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 Ck-data which are generic, in a Baire category sense, 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 rigorous proofs.