REMARKS ON $L^1$ REGULARITY OF GRADIENT FLOWS

A collection of arguments will be presented in order to show that $L^1$ is the natural functional framework for convex gradient flows. In particular, Lipschitz continuous data dependence is not available in general except possibly for the $L^1$ norm.
Examples from solid mechanics and phase transitions illustrate how this information can be exploited for existence and uniqueness of solutions to coupled problems.