Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

TL;DR

This paper investigates the uniqueness, existence, and properties of solutions to nonlocal porous medium type equations, including fractional Laplacian cases, providing theoretical results and analyzing numerical approximations.

Contribution

It establishes general uniqueness and existence results for nonlocal porous medium equations, including new insights into their properties and numerical approximation convergence.

Findings

Proved uniqueness and existence of solutions.

Derived $L^1$-contraction and a priori estimates.

Analyzed numerical approximation convergence.

Abstract

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.