Porous Medium Flow with both a Fractional Potential Pressure and Fractional Time Derivative

TL;DR

This paper investigates a porous medium equation incorporating both a fractional potential pressure and a fractional Caputo time derivative, establishing existence and H"older continuity of weak solutions under certain conditions.

Contribution

It introduces a novel model combining nonlocal diffusion with fractional time derivatives and proves existence and regularity results for weak solutions.

Findings

Existence of weak solutions with exponential decay initial data

H"older continuity of weak solutions

Model captures nonlocal diffusion and memory effects

Abstract

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account "memory''. The precise model is \[ D_t^{\alpha} u - \text{div}(u(-\Delta)^{-\sigma} u) = f, \quad 0<\sigma <1/2. \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0 $ as well as right hand side $f\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is H\"older continuity for such weak solutions.