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

Abstract

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.