Existence of weak solutions to a continuity equation with space time nonlocal Darcy law

Abstract

In this manuscript we consider a porous medium equation with non-local diffusion effects given by a fractional heat operator $\partial_t + (-\Delta)^s$ in two space dimensions. Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality, a higher order integral estimate, and a generalized version of the Div-Curl lemma.