Calculus of Variations and Evolutive Systems on the Wasserstein Space

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This project focuses on the analysis of a collection of variational problems in connection with dynamical and mechanical systems. In particular, it seeks to develop basic tools for studying the calculus of variations and the Monge-Kantorovich theory. In the process of discovering new results in the variational problem realm, it will hopefully unearth new connections with other areas of science and mathematics. In the project, Hamiltonian systems that consist of finitely many particles and possess underlying Poisson structures are considered. When the number of particles becomes infinite, these finite dimensional systems may converge in an appropriate sense to infinite dimensional systems that are encoded as partial differential equations. Much work has been devoted to identifying the Poisson structures for such limiting infinite dimensional systems. Born and Infeld, and independently Pauli, started to develop a quantum field theory in which the commutator operator is analogous to the Poisson bracket studied by Chernoff, Marsden, Weinstein, and many others. In joint work with collaborators, the principal investigator has examined physical systems with no electric or magnetic fields. In this simplified model, they obtained many rigorous results that can be used to handle a class of partial differential equations involving singular measures. Some of the concepts developed by the principal investigator and others are useful in formulating problems such as the formation of coherent structures in connection with the constrained Navier-Stokes equations that has been considered recently by Caglioti, Pulvirenti, and Rousset. The project will investigate these equations and their implications for the two-dimensional Euler equations of incompressible fluids. The ideas that underlie this project are not difficult to explain. Consider a physical system that consists of finitely many particles evolving on a finite-dimensional torus (think of the surface of a doughnut) and assume that the forces applied to the system are derived from a periodic potential. One of the central issues in dynamical system is the search for periodic orbits and so-called invariant measures. In the simple case where there is no force, the periodic orbits and invariant tori can be described explicitly. The celebrated KAM (Kolmogorov-Arnold-Moser) theory ensures that, if the potential is small, then for certain initial conditions one can describe the solutions of the system explicitly in conveniently chosen new coordinates. It is well-known that the existence of these suitable new coordinates is equivalent to the existence of solutions of a "cell problem" that arises in the theory of Hamilton-Jacobi equations. The graph of the latter solution tells one what the "good" initial conditions are. The KAM theory identifies parameters, called rotation vectors, for which smooth solutions (twice-differentiable, say) of the cell problem exist. The "weak" KAM theory considers a larger class of rotation vectors and for each one of them establishes the existence of solutions of the cell problem that are not quite smooth (i.e., that are only "Lipschitz" functions). The principal investigator plans to continue his investigation on the limiting systems where the number of particles becomes infinite. He anticipates his study will shed new light on our understanding of stability issues for partial differential equations. The project will pay special attention to the training of students and the promotion of mathematics in colleges.