  The main goal of this paper is to prove $L^1$-comparison and contraction
principles for weak solutions (in the sense of distributions) of Hele-Shaw flow
with a linear Drift. The flow is considered with a general reaction term
including the Lipschitz continuous case, and subject to mixed homogeneous
boundary conditions : Dirichlet and Neumann. Our approach combines
DiPerna-Lions renormalization type with Kruzhkov device of doubling and
de-doubling variables. The $L^1$-contraction principle allows afterwards to
handle the problem in a general framework of nonlinear semigroup theory in
$L^1,$ taking thus advantage of this strong theory to study existence,
uniqueness, comparison of weak solutions, $L^1$-stability as well as many
further questions.
