The project consists of two main topics: (1) Free boundary problems for elliptic and degenerate elliptic nonlinear equations and systems arising in the models of compressible fluid dynamics. The goal is to study existence, uniqueness and regularity and geometric properties of transonic shocks arising in self-similar shock reflection for potential flows and full compressible Euler system. Euler system for potential flow consists of the conservation law of mass and Bernoulli law for the velocity. In the case self-similar flow the system can be reduced to a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. Transonic shocks are discontinuities in the gradient of a solution, such that the type of equation changes from hyperbolic to elliptic across the shock. Transonic shocks arise in many situations of physical importance. Boundary value problems for transonic shock solutions can be formulated as free boundary problems for the elliptic phase. This framework will be applied to the study of self-similar shock reflection for potential flow. Shock reflection problem will also be considered for full compressible Euler system, which is a more physical model. This leads to the study of free boundary problems for a coupled system consisting of a nonlinear second order equation of mixed elliptic-hyperbolic type and transport equations. One of the goals is to verify von Neumann criteria for transition between regular and Mach reflection. (2) Another area of research is to study the system of semigeostrophic equations, a model of large-scale atmosphere/ocean flows, using methods of Monge-Kantorovich mass transport. In recent years a progress was made in the study of semigeostrophic system with constant Coriolis parameter in the flat geometry. We will study a more physically relevant case of the system with variable Coriolis parameter on a manifold. This includes study of new Monge-Kantorovich-type problems, and Monge-Ampere equations associated with these problems.
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, regularity and geometric properties, 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. Recent progress in Monge-Kantorovich mass transport problem includes many important applications to nonlinear partial differential equations, in particular to the models for front formation in the atmosphere, kinetic theory, fluid flow, elastic crystals, granular materials, and microeconomic decision problems. We plan to work on the applications of mass transportation problem to the models of atmospheric flows. The 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.
First area of the project is the study of free boundary problems arising in the models of compressible fluid and gas dynamics. The PI continued his work on self-similar shock reflection for the potential flow. 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 equations of compressible fluid dynamics. Shocks correspond to discontinuities in the solutions of the governing system of equations. Type of the equation can 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) domain and solution in the elliptic domain, where shock is the free (unknown) boundary. The PI worked on existence and structure of global solutions of the shock reflection problem, their stability, regularity, and geometric properties of free boundaries. An interesting feature of shock reflection problem is the presence of sonic line, where ellipticity of the equation degenerates. Main results obtained in the project are the following: (i) We finalized our work on existence of global solutions of the supersonic regular reflection structure for wedge angles sufficiently close to the ninety degrees; (ii) We worked of extending the global existence result to the wider range of parameters. For relatively weak incident shocks (where the 'weakness' condition is an explicit algebraic condition, and has a geometric interpretation), we showed existence of a global solution for any wedge angle from the maximal interval in which local supersonic regular reflection is possible. For stronger shocks, a qualitatively different reflection pattern may be possible for some parameters (reflected shock attached to the vertex of wedge). In this case, we prove existence of regular reflection solution for wedge angles up to either sonic angle, or up to the shock attachment angle. Technically, a new ingredient is analysis of the geometry of free boundary as the wedge angle changes. (iii) We finalized our work on optimal regularity of the supersonic regular reflection solution near the sonic arc. (iv) We worked on another shock reflection model: Prandtl-Meyer reflection. Obtained existence and optimal regularity of the global solutions of the weak reflection structure for all possible physical parameters. Another area of the project is semigeostrophic system, a model of large-scale atmosphere/ocean flows. The project includes study of semigeostrophic system with singular initial data, which is a physically interesting case. We obtain existence of the appropriately defined weak Lagrangian solutions in the physical space variables. We further show that such solutions conserve the energy naturally related to the model (the geostrophic energy).
