Singular limit of the porous medium equation with a drift

TL;DR

This paper investigates the singular limit of a nonlinear drift-diffusion equation constrained by a maximal density, leading to a Hele-Shaw-type free boundary problem with convergence and regularity results.

Contribution

It introduces a rigorous analysis of the stiff pressure limit with a drift, revealing the emergence of a free boundary problem and establishing convergence and regularity properties.

Findings

Pointwise convergence of densities to the free boundary problem

BV regularity of the limiting free boundary

Identification of the Hele-Shaw-type limit equation

Abstract

We study the "stiff pressure limit" of a nonlinear drift-diffusion equation, where the density is constrained to stay below the maximal value one. The challenge lies in the presence of a drift and the consequent lack of monotonicity in time. In the limit a Hele-Shaw-type free boundary problem emerges, which describes the evolution of the congested zone where density equals one. We discuss pointwise convergence of the densities as well as the $BV$ regularity of the limiting free boundary.