L1-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 of PDE system corresponding to a phase transition diffusion model of Hele-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.