Lectures on Transport Equations and Multi-D Hyperbolic Conservation Laws

Techniques from measure theory, and particularly from geometric measure theory, have been successfully applied in the past few years to study existence, stability, and structure of solutions to nonlinear hyperbolic equations in several space dimensions. The aim of these lectures was to provide an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to the study of hyperbolic conservation laws, in which the use of geometric and measure theoretic tools has played a key role to obtain recent important advances: the well posedness theory of linear transport equations with irregular coefficients, and the study of the BV-like structure of solutions to multi-dimensional scalar conservation laws. There has been also few invited lectures (of one hour each) on topics related to the courses. These courses were attended by over 70 people among Ph.D. students as well as to Post-doctoral and active researchers coming from Italy and four foreign countries: France, Norway, Russia, United States, .