Degenerate diffusion with a drift potential: a viscosity solutions approach

TL;DR

This paper develops a viscosity solutions framework for a nonlinear degenerate diffusion equation with a drift potential, proving convergence of the free boundary to equilibrium and establishing exponential convergence rates for convex potentials.

Contribution

It introduces a new viscosity solutions approach for degenerate diffusion equations with drift, linking it to weak solutions and analyzing free boundary behavior.

Findings

Viscosity solutions coincide with weak solutions.

Free boundary converges uniformly to equilibrium over time.

Exponential convergence rate for convex potentials.

Abstract

We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as time grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.