L1-theory for incompressible limit of reaction-diffusion porous medium flow with linear drift

Our aim is to study existence, uniqueness and the limit, as $m\to \infty$, of the solution of the porous medium equation with linear drift ${\partial }_{t}u-\Delta u^m+\nabla \cdot \left(uV\right)=g(t,x,u)$ in bounded domain with Dirichlet boundary condition. We treat the problem without any sign restriction on the solution with an outpointing vector field V on the boundary and a general source term g (including the continuous Lipschitz case). Under reasonably sharp Sobolev assumptions on V, we show uniform $L^1$-convergence towards the solution of reaction-diffusion Hele-Shaw flow with linear drift.