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.
The PI and his student M. Sedjro developed some techniques for proving the existence and uniqueness of a minimizer in problems where the functional to be optimized is not convex. The current theory of the calculus of variations does not provide us with a dual problem in the case considered in this project, which made the problem very difficult to tackle. The identification of an unusual dual problem was instrumental for achieving their goals. This constitutes the intellectual merit of that project. They applied these techniques to a functional appearing in the study of axisymmetric flows which arise in atmospheric sciences. This constitutes a broader impact. The PI and collaborators are establishing existence of solutions in a Hamiltonian system in which there is a loss of mass. They have introduced some natural distances, useful to prove convergence properties in their work. They recently improved their result by establishing a kind or mass conservation to justify their regularization method. In general in an infinite dimensional setting, such as a Hilbert space, one does not know how to establish existence of solutions for first order PDEs for which the entropy functional is not Lipschitz. Indeed, Picard iteration fails in that case. For a gradient flow with a convex entropy, some existence results can be obtained. Motivated by applications in PDEs and Fluids Mechanics, the PI and collaborators considered the case where the Hilbert space is replaced by a specific infinite dimensional space, the Wasserstein space. They were able to extend De Giorgi interpolation method to PDEs which are not gradient flow on a metric space but possess an underlying Lagrangian. Only the last portion of this project was done under the current contract. The PI and collaborators proved existence of solution satisfying a semi-group property for a sticky particle dynamics with interactions. The interest beyond the mathematical community constitutes the broader impact. In a joint work, the PI and a collaborator extended the weak KAM theory to PDEs such as the one-dimensional Vlasov-Poisson system. The intellectual merit is their recent substantial advance which includes PDEs in higher dimensional spaces. With his PhD student R. Awi, the PI has obtained some uniqueness results on functional involving polyconvex integrand which defy the current state-of-art. This is a progress in the extremely difficult field of the calculus of variations when one deals with quasi convex functions which are not convex.
