Non-Local Porous Media Equations with Fractional Time Derivative

TL;DR

This paper establishes the existence of solutions for a fractional time derivative porous media system with nonlocal operators, using novel approximation and compactness techniques.

Contribution

It introduces a new approach to prove global existence of solutions for fractional nonlocal porous media equations with memory effects.

Findings

Proved global existence of nonnegative weak solutions.

Developed properties and convergence results for the discrete Caputo derivative.

Established a compactness criterion for fractional time derivatives.

Abstract

In this paper we investigate existence of solutions for the system: \begin{equation*} \left\{ \begin{array}{l} D^{\alpha}_tu=\textrm{div}(u \nabla p),\\ D^{\alpha}_tp=-(-\Delta)^{s}p+u^{2}, \end{array} \right. \end{equation*} in $\mathbb{T}^3$ for $0< s \leq 1$, and $0< \alpha \le 1$. The term $D^\alpha_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-\Delta)^{s}$ represents the L\'{e}vy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. Most importantly, we give a strong compactness criteria for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative.