Dissipative Evolutions in Metric Spaces: Well-Posedness and Long-Time Behavior

Motivated by the setting of minimax and multispecies coupled gradient flows, we study a novel class of evolution variational inequalities (EVI) driven by bifunctions defined on a subset of a metric space and satisfying a natural dissipativity condition. 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. We prove existence of solutions via an approach which combines dissipativity with a new notion of convexity along suitable 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. This is joint work with Franca Hoffmann and Giuseppe Savaré.