PDEs and time-dependent gradient flows in rough spaces

Abstract

This project has basically two parts. In the first, we approach partial differential equations (PDEs) in rough spaces, such as weighted-Lp spaces (and sum of them), Morrey spaces, pseudomeasure spaces, Besov spaces, Fourier-Besov spaces, Fourier-Besov-Morrey spaces, Herz type spaces, Besov-Morrey spaces, and investigate the issues of existence, uniqueness, symmetries, renormalization, stability and asymptotic behavior of solutions. The second part consists in studying time-dependent gradient flows in metric spaces and looking for a general theory that can be applied to a number of PDEs and connected with the optimal mass transport theory and Wasserstein space. (AU)