Gradient flow formulation and discrete approximation of reaction-cross-diffusion systems with size exclusion

Abstract

We characterize a class of possibly degenerate nonlinear reaction-cross-diffusion systems with size exclusion as (non-metric) gradient flows in continuity-equation format driven by entropy. We then establish a connection between these systems and discrete reaction networks by proving an EDP-type lower limit inequality for the associated gradient systems. In the non-degenerate case, we show a chain rule identity that allows us to identify minimizers of the corresponding energy-dissipation functional as weak solutions to the PDE system. In conjunction with the lower limit, this yields an alternative proof of existence of weak solutions and their approximability by solutions to the proposed discrete ODE systems. Finally, we generalize our results to models including drift terms by extending the established framework to tilted energy landscapes.