Free Boundary Problems, Mass Transfer and Nonlinear Dynamics

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''Free boundary problems, mass transfer and nonlinear dynamics''Project AbstractMikhail FeldmanThe project consists of two main topics: (1) Free boundary problems arising in the models of compressible fluid dynamics. The objective is to study existence, uniqueness and stability of multidimensional transonic shocks for Euler equations for steady and self-similar potential flows. Euler equations, consisting of the conservation law of mass and Bernoulli law for the velocity, can be written as a second-order, nonlinear elliptic-hyperbolic equation for the velocity potential, if the flow is steady or self-similar. Transonic shocks are discontinuities in the gradient of a solution, such that the type of equation changes from hyperbolic to elliptic across the shock surface. Transonic shocks arise in many situations of physical importance. Boundary value problems for transonic shock solutions can be formulated as a free boundary problem for the elliptic phase. This framework will be applied to study several important situations, including self-similar shock reflection, and steady transonic shocks in supersonic flow past conical body and smooth convex obstacle. These problems will also be considered in the more general framework of full compressible Euler system.(2) Another area of research is Monge-Kantorovich mass transfer problem. The questions to study include geometric and measure-theoretic properties of optimal maps, and applications to partial differential equations arising in physical models. This includes the system of semigeostrophic equations, a model of large-scale atmosphere/ocean flows. Another area is to study Monge problem on Heisenberg group.Free boundary problems arise naturally in many models in physics, fluid dynamics, economics. Free boundaries correspond to sharp changes in the variables describing the problem. Significant progress has been made during last several decades in the study of free boundary problems. However in the case of nonlinear partial differential equations and especially equations of mixed type many important questions are yet to be studied. This is the first theme of the project. Better understanding of properties of free boundaries, such as stability, makes possible to understand complex phenomena in models and applications. We plan to study transonic shocks in a flow of compressible fluid or gas. Another area of the project is optimal transportation problem. Recently many fundamental properties and important applications of this problem within and beyond mathematics were discovered, in particular its connections to nonlinear partial differential equations, and applications in models for front formation in the atmosphere, kinetic theory, fluid flow, elastic crystals, granular materials, and microeconomic decision problems. We plan to work on theory and applications of optimal transportation problem. A broader impact resulting of the project will be the analysis of some important problems in engineering and meteorology.