We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply these ideas to identify a model of animal dispersal proposed by MacCall and Cosner as a gradient flow in our formalism and obtain new long-time convergence results.

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.

We consider a fitness-driven model of dispersal of N interacting populations, which was previously studied merely in the case N = 1. Based on some optimal transport distance recently introduced, we identify the model as a gradient flow in the metric space of Radon measures. We prove existence of global non-negative weak solutions to the corresponding system of parabolic PDEs, which involves degenerate cross-diffusion. Under some additional hypotheses and using a new multicomponent Poincaré-Beckner functional inequality, we show that the solutions converge exponentially to an ideal free distribution in the long time regime.

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We study the dynamics of the Jeffreys-Oldroyd equation using the theory of trajectory pullback attractors. We prove an existence theorem for weak solutions and use it to construct a family of trajectory spaces and to specify the class of attracted families of sets, which includes families bounded in the past. Finally, we prove the existence of the trajectory and global pullback attractors of the model.