Measure-valued and discontinuous solutions of some evolution equations

Motivated by an application to stratified turbulent flows in oceanography, I shall discuss some singular solutions of nonlinear PDE's of evolutionary type, in particular discontinuous and Radon measure valued solutions. We focus on different regularizations of backward parabolic equations, first order scalar conservation laws, and, if time permits, a toy problem for a Hamilton-Jacobi equation. Most of this lecture is based on collaborations with L. Giacomelli, F. Smarrazzo, A. Terracina and A. Tesei.