Measure valued solutions of sub-linear diffusion equations with a drift term

TL;DR

This paper investigates measure-valued solutions to one-dimensional drift-diffusion equations with degenerate nonlinear diffusion, revealing how solutions behave over time depending on a critical mass and utilizing optimal transport tools.

Contribution

It introduces a well-posedness framework for measure solutions with finite mass and momentum, characterizes large-time behavior, and identifies the impact of critical mass on solution regularity.

Findings

Solutions are well-posed in the measure space with finite mass and momentum.

Large-time behavior depends on a critical mass ${m}_{ m c}$, with subcritical masses leading to absolutely continuous steady states.

Supercritical masses result in steady states with singular parts where excess mass accumulates.

Abstract

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.