Convergence rate for the incompressible limit of nonlinear diffusion-advection equations

TL;DR

This paper investigates the rate at which solutions to nonlinear diffusion equations approach the incompressible limit, providing quantitative convergence estimates in Sobolev and Lebesgue spaces.

Contribution

It introduces the first known convergence rate estimates for solutions of nonlinear diffusion-advection equations approaching the incompressible limit.

Findings

Convergence rate computed in negative Sobolev norm.

Interpolated convergence rate in Lebesgue spaces.

Provides quantitative bounds on the incompressible limit.

Abstract

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cell-population models to free boundary problems of Hele-Shaw type. Although vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.