Convergence and nonconvergence in a nonlocal gradient flow
Year: 2024
Type: article
Abstract: We study the asymptotic convergence as of solutions of , a nonlocal differential equation that is formally a gradient flow in a constant‐mass subspace of 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 Łojasiewicz‐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 . Curiously, the exponential rate of convergence in the finite‐value case can jump from order to arbitrarily small values upon perturbation of parameters.
DOI: 10.1112/jlms.70047
Authors: Sangmin Park, Robert L. Pego
Institution: Carnegie Mellon University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,780
Funder: National Science Foundation
Awards: Variational Problems and Partial Differential Equations on Discrete Random Structures: Analysis and Applications to Data Science, Collaborative Research: Dynamics, singularities, and variational structure in models of fluids and clustering, Novel Transportation-Based Geometries, Gradient Flows, and Applications to Data Science
OpenAlex ID: W4405290348