Convergence and non-convergence in a nonlocal gradient flow

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal gradient flow equation, establishing conditions for convergence or divergence, and revealing how small parameter changes can significantly affect convergence rates.

Contribution

It provides a new proof of stabilization for solutions with finitely many values and demonstrates non-convergence in cases with infinitely many values, highlighting the impact of perturbations.

Findings

Solutions with finitely many values converge to equilibrium with a rate that can vary widely.

Solutions with infinitely many values may not converge, as shown by counterexamples.

Small parameter perturbations can cause the convergence rate to change dramatically.

Abstract

We study the asymptotic convergence of solutions as $t\rightarrow\infty$ of $\partial_t u=-f(u)+\int f(u)$, a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of $L^2$ arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a {\L}ojasiewicz-type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in general need not converge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise linear and cubic functions $f$. Curiously, the exponential rate of convergence in the finite-value case can jump from order $O(1)$ to arbitrarily small values upon perturbation of parameters.