Non-local porous media equations with fractional time derivative. (October 2021)

• Record Type: Journal Article
• Authors: Daus, Esther; Gualdani, Maria Pia; Xu, Jingjing; Zamponi, Nicola; Zhang, Xinyu
• Is Part Of: Nonlinear analysis. Volume 211(2021)
• Publication Date: 2021-10
• Subjects: Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112486
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we investigate existence of solutions for the system: D t α u = div ( u ∇ p ), D t α p = − ( − Δ ) s p + u 2, in T 3 for 0 < s ≤ 1, and 0 < α ≤ 1 . The term D t α 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 strong compactness criteria for piecewise constant functions, in the spirit of Aubin–Lions theorem, based on bounds of the discrete Caputo derivative.