Free Boundary Problems and Mass Transfer

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

PI: Mikhail Feldman, University of Wisconsin DMS-0200644 ABSTRACT -------------------------------------------------- The project consists of two main topics: (1) Free boundary problems for nonlinear elliptic equations. One of objectives is to develop techniques for studying free boundary problems arising in the models of compressible fluid dynamics, in particular to study existence, uniqueness and stability of multidimensional transonic shocks for Euler equations for steady and self-similar potential flows. Euler equations can be written as a single second order, nonlinear elliptic-hyperbolic equation of mixed type for the velocity potential, if the flow is steady or self-similar. Transonic shocks are discontinuities in the gradient of the 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 (steady supersonic flows around an obstacle, shock reflection for self-similar flows). (2) Monge-Kantorovich mass transfer problem. The questions to study include geometric and measure-theoretic properties of solutions, and applications to partial differential equations. 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 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.