Drift-diffusion equations with saturation

TL;DR

This paper investigates nonlinear saturation drift-diffusion equations with constrained densities, establishing existence, long-term behavior, free boundary phenomena, and a structure-preserving numerical scheme.

Contribution

It introduces a framework for analyzing saturation problems with non-concave mobility, proving existence of solutions, studying their long-term dynamics, and developing a convergent numerical scheme.

Findings

Existence of $C_0$-semigroups of $L^1$ contractions for the equations.

Characterization of long-time behavior and free boundary formation.

Development and analysis of a structure-preserving finite-volume scheme.

Abstract

We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $\rho$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(\rho)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour.