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

TL;DR

This paper proves the global existence of weak solutions for a porous medium equation with non-local fractional diffusion effects in two dimensions, using semi-discretization, energy estimates, and advanced mathematical tools.

Contribution

It introduces a novel approach to establish global weak solutions for a fractional porous medium equation with non-local effects in two spatial dimensions.

Findings

Established global existence of weak solutions

Developed new energy and integral estimates

Applied generalized Div-Curl lemma in non-local PDE context

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.