Uniform regularity estimates for nonlinear diffusion-advection equations in the hard-congestion limit

TL;DR

This paper establishes regularity estimates for nonlinear diffusion-advection equations of porous medium type, including their incompressible limit, by relaxing drift assumptions and analyzing the impact of linear diffusion on pressure regularity.

Contribution

It introduces new regularity results for porous medium equations with drifts, extending previous work by relaxing assumptions and studying linear diffusion effects.

Findings

Pressure gradient in porous medium flows is $L^4$-summable and stable across nonlinearities.

Pressure Hessian estimates are obtained in $L^2$, with pressure being the positive part of an $H^2$-function in the linear diffusion case.

Regularity results hold under relaxed drift assumptions and include the incompressible limit scenario.

Abstract

We present regularity results for nonlinear drift-diffusion equations of porous medium type (together with their incompressible limit). We relax the assumptions imposed on the drift term with respect to previous results and additionally study the effect of linear diffusion on our regularity result (a scenario of particular interest in the incompressible case, for it represents the motion of particles driven by a Brownian motion subject to a density constraint). Specifically, this work concerns the $L^4$-summability of the pressure gradient in porous medium flows with drifts that is stable with respect to the exponent of the nonlinearity, and $L^2$-estimates on the pressure Hessian (in particular, in the incompressible case with linear diffusion we prove that the pressure is the positive part of an $H^2$-function).