SBV Regularity Results for Solutions to 1D Conservation LawsHyperbolic Conservation Laws and Related Analysis with Applications

A well-posedness theory has been established for entropy solutions to strictly hyperbolic systems of conservation laws, in one space variable, with small total variation. We give in this note an introduction to SBV-regularity results: when the characteristic fields are genuinely nonlinear, the derivative of an entropy solution consists only of the absolutely continuous part and of the jump part, while a fractal behavior (the Cantor part) is ruled out. We first review the scalar uniformly convex case, related to the Hopf-Lax formula. We then turn to the case of systems: one has a decay estimate for both positive and negative waves, obtained considering the interaction-cancellation measures and balance measures for the jump part of the waves. When the Cantor part of the time restriction of the entropy solution does not vanish, either the Glimm functional has a downward jump, or there is a cancellation of waves or this wave balance measure is positive, and this can occur at most at countably many times. We then remove the assumption of genuine nonlinearity. The Cantor part is in general present. There are however interesting nonlinear functions of the entropy solution which still enjoy this regularity.