Dissipative Evolutions in Metric Spaces

Abstract

We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ defined on a subset $\mathrm{D}$ of a metric space $(\mathrm{X},\mathrm{d})$ and satisfying a natural dissipativity condition $$\mathsf{b}(x,y)+\mathsf{b}(y,x)\le η\mathrm{d}^2(x,y).$$ We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\mathsf{b}$ with a new notion of convexity along suitable $\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.