Existence of solutions to reaction cross diffusion systems

Abstract

Reaction cross diffusion systems are a two species generalization of the porous media equation. These systems play an important role in the mechanical modeling of living tissues and tumor growth. Due to their mixed parabolic-hyperbolic structure, even proving the existence of solutions to these equations is challenging. In this paper, we exploit the parabolic structure of the system to prove the strong compactness of the pressure gradient in L2. The key ingredient is the energy dissipation relation, which along with some compensated compactness arguments, allows us to upgrade weak convergence to strong convergence. As a consequence of the pressure compactness, we are able to prove the existence of solutions in a very general setting and pass to the Hele-Shaw/incompressible limit in any dimension.