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The main goal of this paper is to prove $L^1$-comparison and contractionprinciples for weak solutions (in the sense of distributions) of Hele-Shaw flowwith a linear Drift. The flow is considered with a general reaction termincluding the Lipschitz continuous case, and subject to mixed homogeneousboundary conditions : Dirichlet and Neumann. Our approach combinesDiPerna-Lions renormalization type with Kruzhkov device of doubling andde-doubling variables. The $L^1$-contraction principle allows afterwards tohandle 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 manyfurther questions.

$L^1$-Theory for Hele-Shaw flow with linear drift