Evolution Problems and Free Boundaries

该项目包括两个主要主题：(1)可压缩流体动力学模型中出现的椭圆和退化椭圆非线性方程和系统的自由边界问题。目的是研究势流和全可压缩欧拉系统自相似激波反射中跨声速激波的存在性、唯一性、规律性和几何性质。势流的欧拉系统由质量守恒定律和速度的伯努利定律组成。在自相似流的情况下，系统的速度势可以简化为二阶混合椭圆-双曲型非线性方程。跨音速激波是溶液梯度中的不连续，使得方程的类型在激波上从双曲型变为椭圆型。跨音速冲击出现在许多重要的物理场合。跨声速激波解的边值问题可以表述为椭圆相的自由边值问题。该框架将应用于势流自相似激波反射的研究。对于完全可压缩欧拉系统，也将考虑冲击反射问题，这是一个更物理的模型。这导致了对由混合椭圆-双曲型非线性二阶方程和输运方程组成的耦合系统的自由边界问题的研究。其中一个目标是验证在正则反射和马赫反射之间转换的冯·诺依曼准则。(2)另一个研究领域是利用Monge-Kantorovich质量输运方法研究大尺度大气/海洋流动模式——半顺转方程组。近年来，平面几何中具有恒定科里奥利参数的半转系统的研究取得了进展。我们将研究一个在流形上具有可变科里奥利参数的系统的更为物理相关的情况。这包括研究新的monge - kantorovich型问题，以及与这些问题相关的Monge-Ampere方程。自由边界问题在物理学、流体动力学、经济学的许多模型中自然出现。自由边界对应于描述问题的变量的急剧变化。近几十年来，自由边界问题的研究取得了重大进展。但是对于非线性偏微分方程，特别是混合型偏微分方程，还有许多重要的问题有待研究。这是该项目的第一个主题。更好地理解自由边界的性质，如稳定性、规律性和几何性质，使理解模型和应用中的复杂现象成为可能。我们计划研究可压缩流体或气体流动中的跨音速激波。项目的另一个领域是最优运输问题。monger - kantorovich质量输运问题的最新进展包括许多非线性偏微分方程的重要应用，特别是在大气锋面形成模型，动力学理论，流体流动，弹性晶体，颗粒材料和微观经济决策问题。我们计划研究质量输运问题在大气流动模型中的应用。由于该项目解决了工程和气象学中的重要问题，因此该项目将产生更广泛的影响。此外，研究生将参与该项目的工作。

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.