High-frequency limit of non-autonomous gradient flows

TL;DR

This paper investigates how non-autonomous gradient flows with rapidly oscillating time-dependent perturbations behave in the high-frequency limit, showing convergence to a time-averaged evolution and providing explicit convergence rates under certain convexity conditions.

Contribution

It extends existing existence results to geodesically non-convex energies and establishes convergence of solutions to a time-averaged problem, including explicit rates under geodesic convexity.

Findings

Solutions converge to a time-averaged evolution as frequency increases.

Explicit convergence rates are derived under geodesic λ-convexity.

Weak solutions of nonlinear drift-diffusion equations exhibit the predicted high-frequency behavior.

Abstract

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way, but fulfills a certain Lipschitz condition. On grounds of the existence results by Ferreira and Guevara (2015) on non-autonomous gradient flows (which we also extend to the framework of geodesically non-convex energies), we prove that the associated solution curves converge to a solution of the time-averaged evolution equation in the limit of infinite frequency. Under the additional assumption of dynamical geodesic $\lambda$-convexity of the energy, we obtain an explicit rate of convergence. In the non-convex case, we specifically investigate nonlinear drift-diffusion equations with time-dependent drift which are gradient flows with respect to the $L^2$-Wasserstein distance. We prove that a family of weak solutions obtained as a limit of the Minimizing Movement scheme exhibits the above-mentioned behaviour in the high-frequency limit.