L 1 -THEORY FOR 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 of PDE system corresponding to a phase transition diffusion model ofHele-Shaw type with addition of a linear drift. The flow is considered with a source term and subject to mixed homogeneous boundary conditions : Dirichlet and Neumann. The PDE can be focused to model for instance biological applications including multi-species diffusion-aggregation models and pedestrian dynamics with congestion. 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.
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