Nonlinear free boundary and evolution problems

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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. Better understanding of properties of free boundaries, such as stability, regularity and geometric structure, makes possible to study complex phenomena in models and applications. First area of this project is the study of shock reflection problem in gas dynamics, one of the most fundamental multidimensional shock wave problems. This study involves several challenging difficulties in analysis of free boundary problems for nonlinear partial differential equations. Another area of project is semigeostrophic system, a model of rotation-dominated atmospheric/ocean flows. It exhibits a rich mathematical structure based on Monge-Kantorovich mass transport theory. We plan to study physically realistic case of variable Coriolis parameter in semigeostrophic model. Broader impact resulting from the project will be achieved since the project addresses fundamental mathematical models in engineering and atmospheric sciences. Closer interaction with engineering and meteorological communities is one of the priorities of the project. Graduate students will be involved in the project.The project consists of two main topics: (1) Free boundary problems in shock analysis. The PI will continue his work on self-similar shock reflection for potential flow and for full and isentropic Euler system. Shock reflection problems arise in many physical situations. Moreover, such problems are important in the mathematical theory of multidimensional conservation laws since their solutions are building blocks and asymptotic attractors of general solutions to the multidimensional Euler equations for compressible fluids. Self-similar equations of compressible fluid dynamics are of mixed elliptic-hyperbolic type. Shocks correspond to discontinuities in the solution for Euler system, and in the gradient of the solution for potential flow equation. Type of equation may change from hyperbolic to elliptic across the shock. Shock reflection problem can be formulated as a free boundary problem in which unknown are the elliptic (subsonic) region and solution in the elliptic region. The PI will continue his work on existence, stability and regularity of global solutions of the regular reflection, to extend the global existence results to the case of compressible Euler system, which is a fundamental model of gas dynamics. Further study includes uniqueness and stability for regular reflection problem. (2) Another area of the proposed research is semigeostrophic system. The PI will study semigeostrophic system with variable Coriolis parameter on a Riemannian manifold. Such model arises from taking into account the curvature of the Earth. The PI also plans study solutions with low regularity, which come from modeling of flows with neutrally stable regions. These projects involve study of Monge-Kantorovich mass transport problems, and Lagrangian solutions of transport equations with vector fields of low regularity.