L1-Theory for Hele-Shaw flow with linear drift

Abstract

The main goal of this paper is to prove [Formula: see text]-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 [Formula: see text]-contraction principle allows afterwards to handle the problem in a general framework of nonlinear semigroup theory in [Formula: see text], thus taking advantage of this strong theory to study furthermore existence, uniqueness, comparison of weak solutions, [Formula: see text]-stability as well as many further questions.