The project consists of two main topics: (1) a study of free boundary problems for elliptic and degenerate elliptic nonlinear equations and systems that arise in the models of compressible fluid dynamics and (2) a study of the system of semigeostrophic equations, which provides a model of large-scale atmosphere/ocean flows, using methods of Monge-Kantorovich mass transport and transport equations with nonsmooth vector fields. The first part of the project focuses on the shock reflection problem, one of the most fundamental multidimensional shock wave problems. The objective is to study existence, uniqueness, and geometric properties of solutions to the self-similar potential flow equation and the full compressible Euler system, which together model regular shock reflection. The self-similar potential flow equation is a nonlinear second-order equation of mixed elliptic-hyperbolic type for the velocity potential. The regular shock reflection problem can be formulated as a free boundary problem for the elliptic phase of the solution. In recent work, G.-Q. Chen and the principal investigator have established the global existence of a regular shock reflection solution for the potential flow in the case where the wedge angle is larger than the sonic angle. The goal of the present project is to extend these results in several directions, including proving existence of subsonic regular reflection (thus completing the proof of the von Neumann detachment conjecture in the framework of potential flow) and including the case of the compressible Euler system. In recent years, progress has been made in the study of the semigeostrophic system with constant Coriolis parameter in flat geometry. The second component of this project will include the investigation of a more physically relevant case of the system, one with variable Coriolis parameter on a manifold, and also a study of solutions that correspond to singular measures in the "dual" variables.
Free boundary problems arise in many models in physics, fluid dynamics, engineering, and economics. In physical systems, "free boundaries" are regions of rapid variation of conditions between two very different states, such as shock wave in gas dynamics. Mathematically this rapid transition is simplified as occurring infinitely fast along a surface of discontinuity in the partial differential equation governing the physics. Location of this surface is not known at the outset, thus one must solve both for physical states and their boundaries. Significant progress has been achieved during the last several decades in the study of free boundary problems. However, in the case of nonlinear partial differential equations and especially for equations that have very different properties in the regions separated by the free boundary, many questions remain. The principal investigator plans to apply the techniques of free boundary problems to study some fundamental multidimensional shock waves in gas dynamics, specifically shock reflection patterns. This involves free boundary problems for nonlinear equations and systems of complex structure, for which new methods will be needed. Understanding properties of free boundaries, such as their regularity, stability and geometry, allows better analysis and numerical methods in models and applications. Another focus of the project is the semigeostrophic system, which models large-scale atmospheric/oceanic flows and is used in meteorology, in particular in models of front formation in the atmosphere. Methods include study of related Monge-Kantorovich-type problems. The Monge-Kantorovich mass transport theory has recently been applied successfully in several areas (e.g., kinetic theory, fluid flow, elastic crystals, granular materials, urban planning, microeconomic decision problems). Broader impact resulting from the project will be achieved since the project addresses the problems important in engineering and meteorology. Also, graduate students will be involved in the work on the project.
