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 general isoperimetric-type functional 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.

Spherical Hellinger--Kantorovich Gradient Flows

“…Under our assumptions, the entropy, generally speaking, possesses neither geodesic convexity nor semi-convexity with respect to either the spherical or conic Hellinger-Kantorovich structure, or even to the classical Wasserstein one, cf. [32,30]. Remark 1.3.…”

“…The fitness manifests itself as a growth rate, and simultaneously affects the dispersal as the species move along its gradient towards the most favorable environment. In terms of the PDEs, this can be expressed [32] in the following manner:…”

“…The direct dependence on x expresses the spatial inhomogeneity of the resources. The dependence on the normalized population density (in contrast with [36,14,15,16,32] and the references therein, where the fitness depends on the density U itself) models the phenomenon that the individuals compare the quality of their life with the ones of the other members of the society, and their fitness is determined by their relative success in comparison with the others. This model seems to be specifically relevant for those human societies where the population growth (which depends on various factors including fertility, ability of children to survive, longevity etc.)…”

“…The problem (1.12)-(1.14) resembles a conic Hellinger-Kantorovich gradient flow, cf. [32], but this guess is wrong. The reason is that (1.15) is not an L 2 variation of any functional.…”

“…However, the last implication is only valid for probability distributions u, whereas (1.26) would not be a consequence of (1.23) for u of arbitrary mass. These three inequalities can be used to derive exponential convergence to the equilibrium m for the corresponding gradient flows, see [48,49,32] as well as Theorems 3.9 and 3.12 below.…”