Abstract

The overall aim of the course is to review a number of recent results illustrating the great variety of dynamic behavior possible in infinite-dimensional problems, going into some detail on several of the key aspects.

Convergence and nonconvergence of gradient flows: Gradient-flow structure is widely investigated and is nowadays of great importance in optimization and machine learning. Despite having finite dissipation, gradient flows can fail to converge in large time. We’ll discuss criteria that can ensure convergence, including Lojasiewicz gradient estimates, surprising examples showing how convergence can fail in infinite dimensions, and some open problems. Models discussed will come mainly from population biology and phase transition dynamics.

Optimal breaking flows and Minkowski’s problem for polytopes: Least-action principles for dynamics have seen much activity in the last few decades related to geodesic flows of diffeomorphisms and Wasserstein distance. Motivated by results relating relaxation of action minimization for free-boundary incompressible flows to Wasserstein distance, we study optimizing incompressible flows. Such flows are locally rigid and characterized by a countable Alexandrov theorem related to Minkowski’s study of convex bodies and Gauss curvature. Examples and conjectures will be discussed.

Long waves and solitons in lattices with forces of infinite range: Solitary waves have been known to exist in particle lattices with nearest-neighbor forces since the 1994 work of Friesecke and Wattis.
Small-amplitude, near-sonic waves are well described by the KdV equation in lattices with finite-range forces. For infinite-range, power-law forces that decay slower than the fourth power of distance, nonlocal effects appear in the continuum limit. We describe this through formal asymptotics, describe remarkable formulas for solitons in the inverse-cube case (Calogero-Moser), and discuss a new existence theorem for near-sonic solitary waves in this regime.