$L^1-$Theory for Incompressible Limit of Reaction-Diffusion Porous Medium Flow with Linear Drift

Abstract

Our aim is to study the limit of the solution of reaction-diffusion porous medium equation with linear drift ${\partial }_{t}u-\Delta u^m+\nabla \cdot (uV)=g(t,x,u)$, as $m\to \infty .$ We study the problem in bounded domain $\Omega$ with Dirichlet boundary condition, compatible initial data ; i.e. $|u_0|\le 1,$ and an outpointing vector field $V$ on the boundary $\partial \Omega .$ In particular, by means of new $B{V}_{loc}$ estimates, we show uniform $L^1-$convergence towards the solution of reaction-diffusion Hele-Shaw flow with linear drift.