Non-Local Porous Media Equations with Fractional Time Derivative

2020/10/30 by Esther S. Daus, Maria Pia Gualdani, Daus, Esther S. +7
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2010.16332

openalex publication_date 2020/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we investigate existence of solutions for the system: \ D^α_tu=\textrmdiv(u ∇ p),
D^α_tp=-(-Δ)^sp+u², . in \mathbbT³ for 0< s ≤ 1, and 0< α≤ 1. The term D^α_t u denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian (-Δ)^s represents the Lévy 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.

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