L 1 -THEORY FOR REACTION-DIFFUSION HELE-SHAW FLOW WITH LINEAR DRIFT
Résumé
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 a renormalization proceedings à la DiPerna-Lions with Kruzhkov device of doubling and de-doubling variables. The L 1contraction principle allows 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.
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