Monotone Multispecies Flows

Abstract: We present a novel notion of $\lambda$-monotonicity for an $n$-species system of partial differential equations governed by mass-preserving flow dynamics, extending monotonicity in Banach spaces to the Wasserstein-2 metric space. We show that monotonicity implies the existence of and convergence to a unique steady state, convergence of the velocity fields and second moments, and contraction in the Wasserstein-2 metric, at rates dependent on $\lambda$. In the special setting of Wasserstein-2 gradient descent of different energies for each species, we prove convergence to the unique Nash equilibrium of the associated energies and delineate the relationship between monotonicity and displacement convexity. This extends known zero-sum results in infinite-dimensional game theory to the general-sum setting. We provide a number of examples of monotone coupled gradient flow systems, including cross-diffusion, gradient flows with potentials, nonlocal interaction, linear and nonlinear diffusion, and min-max systems, and draw connections to a class of mean-field games. Numerically, we demonstrate convergence of a four-player economic model for service providers and strategic users competing in a market, and a degenerately monotone game.