Degenerate diffusion with a drift potential: a viscosity solutions approach, joint work with I. C. Kim, truncated version

TL;DR

This paper introduces a viscosity solutions framework for a degenerate diffusion equation with a drift potential, demonstrating free boundary convergence to equilibrium in the convex case, while clarifying previous results and removing certain claims.

Contribution

It develops a viscosity solutions approach for degenerate diffusion with a drift potential and establishes free boundary convergence in the convex potential case.

Findings

Viscosity solutions coincide with weak solutions for the equation.

Free boundary converges uniformly to equilibrium with convex potential.

Removed claims of quantitative convergence rates from previous versions.

Abstract

This is a truncated version of the paper "Degenerate diffusion with a drift potential: a viscosity solutions approach", co-authored with I. C. Kim. The purpose of this version is to withdraw the claim of quantitative rate of convergence of the free boundary on the part of H. K. Lei. The difference from the previous version lies in Section 3 where 1) the quantitative version of the convergence of the free boundary statement has been removed and 2) the more basic version of some convergence of the free boundary given uniform convergence of the function has been rewritten. It is emphasized that while some effort has been made towards better exposition and clarity with regard to showing some convergence of the free boundary given uniform convergence of the function (see Section 3) there is no new result here. Quite on the contrary, as the title indicates, what is contained here is a strict subset of the original. We introduce a notion of viscosity solution for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solution coincides with that of the weak solu- tion defined via integration by parts. As an application of the viscosity solutions theory, we show that in the case of a strictly convex potential, the free boundary uniformly converges to equilibrium as $t$ grows.