Calculus of Variations and Hamilton-Jacobi Equations in the Wasserstein Space

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project aims to study a class of dynamical systems on the Wasserstein space of probability measures corresponding to some fundamental systems of partial differential equations in fluid and quantum mechanics. A nice feature of the Wasserstein space approach is its ability to handle singular data and singular solutions. In addition, there is also a possibility of handling discrete and continuous models with the same formalism. The core of this program lies in the development of an appropriate theory for calculus of variations and Hamilton-Jacobi equations in the Wasserstein space. In this project, the principal investigator will investigate the following topics: minimizing the action functional for certain physically meaningful Lagrangians defined on the set of absolutely continuous paths in the Wasserstein space of probability measures; existence and uniqueness of viscosity solutions for Hamilton-Jacobi equations in the Wasserstein space; and homogenization of these Hamilton-Jacobi equations. Some of the most challenging aspects to these problems are the lack of local compactness in the space of probability measures and the presence of strong singularities in the Lagrangians and Hamiltonians.Calculus of variations and Hamilton-Jacobi equations play a central role in many areas of sciences including physics, economics, and engineering. In particular, they are the main object of study in classical mechanics. It is expected that the properties of solutions of the variational problems and the Hamilton-Jacobi equations to be studied in this project will provide a significant step in understanding the dynamics of some infinite-dimensional dynamical systems arising in fluid and quantum mechanics. The principal investigator's Wasserstein formalism approach is also expected to give a unifying framework in which both classical and quantum behavior of particles can be described. The proposed activity will provide a fruitful interaction with some other disciplines of theoretical and applied mathematics such as optimal mass transportation, hyperbolic systems of conservation laws, probability theory, and infinite-dimensional optimal control theory.