Singularly Perturbed Gradient Flows and Evolution of Critical Points in Infinite Dimensions

TL;DR

This paper studies the behavior of singularly perturbed gradient flows in infinite-dimensional spaces, establishing convergence to critical point curves and introducing new solution concepts that capture dissipation and viscous effects.

Contribution

It develops novel compactness and convergence techniques for gradient flows with nonconvex, nonsmooth energies, and introduces Dissipative and Balanced Viscosity solutions with detailed dissipation measures.

Findings

Subsequential limits are Dissipative Viscosity solutions.

Under certain conditions, solutions improve to Balanced Viscosity solutions with atomic dissipation.

Kernel dimension conditions relate to transversality and are generically satisfied.

Abstract

We consider singularly perturbed gradient flows in Hilbert spaces, driven by a time-dependent, nonconvex, and nonsmooth energy, and address the convergence of their solutions to curves of critical points of the driving energy functional. The degenerating nature of the estimates along the gradient-flow curves calls for novel compactness arguments, which we carefully develop by combining tools from the variational approach to Hilbert and metric gradient flows \cite{RossiSavare06,AGS08}, with fine requirements on the set of critical points of the energy. This leads us to prove that subsequential limits of singularly perturbed gradient flows are Dissipative Viscosity solutions of the limiting problem, i.e., a curves of critical points satisfying a suitable balance between the energy and a defect measure, encoding dissipation. This energy-dissipation balance encompasses information on the dynamics of the process at jump times, recording, in particular, the re-emergence of viscous behavior. Under a suitable rectifiability condition on the critical set, we show that Dissipative Viscosity solutions improve to Balanced Viscosity solutions, which have the key property that the dissipation measure is purely atomic. In the second part of the paper we show that, for smooth energies whose second differential is a Fredholm operator, the condition that the kernel of the Hessian has dimension at most one at every critical point already implies our measure-theoretic assumptions. We further relate them to the transversality conditions from bifurcation theory and show that they have a generic character.