  Our aim is to study the limit of the solution of reaction-diffusion porous
medium equation with linear drift $\displaystyle\partial_t u -\Delta u^m
+\nabla \cdot (u \: V)=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. $\vert u_0\vert \leq 1,$ and an outpointing vector field $V$ on the
boundary $\partial \Omega.$ In particular, by means of new $BV_{loc}$
estimates, we show uniform $L^1-$convergence towards the solution of
reaction-diffusion Hele-Shaw flow with linear drift.
