Nonlinear Fokker-Planck equations with reaction as gradient flows of the free energy

Abstract

We interpret a class of nonlinear Fokker-Planck equations with reaction as gradient flows over the space of Radon measures equipped with the recently introduced Hellinger-Kantorovich distance. The driving entropy of the gradient flow is not assumed to be geodesically convex or semi-convex. We prove new generalized dissipation inequalities, which allow us to control the relative entropy by its production. We establish the entropic exponential convergence of the trajectories of the flow to the equilibrium. Along with other applications, this result has an ecological interpretation as a trend to the ideal free distribution for a class of fitness-driven models of population dynamics. Our existence theorem for weak solutions under mild assumptions on the nonlinearity is new even in the absence of the reaction term.