Abstract:
This course explores the structure and regularity of bounded entropy solutions to scalar conservation laws. While such solutions do not generally possess bounded variation (BV), they are expected to share several fine properties with BV functions. We specifically address the rectifiability of entropy dissipation measures, providing positive answers in particular situations. Finally, we connect these regularity properties to open problems regarding the singular limit of phase transition models (such as the Aviles-Giga functionals), where energy is expected to concentrate on co-dimension 1 interfaces.