Dissipative Evolutions in Metric Spaces
Abstract
We study a novel class of evolution variational inequalities (EVI) driven by bifunctions b:D × D → ℝ defined on a subset D of a metric space (X,d) and satisfying a natural dissipativity condition b(x,y)+b(y,x)≤ ηd2(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 b 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.
Related
BibTeX
@misc{conger2026dissipative,
title = {{Dissipative Evolutions in Metric Spaces}},
author = {Lauren Conger and Franca Hoffmann and Giuseppe Savaré},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20998},
eprint = {2609.20998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20998},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dissipative Evolutions in Metric Spaces
AU - Lauren Conger
AU - Franca Hoffmann
AU - Giuseppe Savaré
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20998
UR - https://arxiv.org/abs/2609.20998
AB - We study a novel class of evolution variational inequalities (EVI) driven by bifunctions b:D × D → ℝ defined on a subset D of a metric space (X,d) and satisfying a natural dissipativity condition b(x,y)+b(y,x)≤ ηd^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 b 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.
ER -