Free boundary and evolution problems arising in gas dynamics

本项目包括两个主要课题：(1)研究可压缩流体动力学模型中出现的椭圆型和退化椭圆型非线性方程和系统的自由边界问题；(2)研究半顺转方程系统，该系统利用Monge-Kantorovich质量输运和非光滑矢量场输运方程的方法提供了大尺度大气/海洋流动的模型。项目的第一部分重点研究了最基本的多维激波问题之一——冲击反射问题。目的是研究共同模拟规则激波反射的自相似势流方程和全可压缩欧拉系统解的存在唯一性和几何性质。自相似势流方程是速度势的二阶混合椭圆-双曲型非线性方程。正则激波反射问题可以表示为解的椭圆相的自由边界问题。在最近的研究中，g - q。Chen和主要研究者已经建立了楔形角大于声角的势流的规则激波反射解的全局存在性。本项目的目标是在几个方向上扩展这些结果，包括证明亚音速规则反射的存在性（从而完成在势流框架下的冯·诺伊曼分离猜想的证明），并包括可压缩欧拉系统的情况。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一定的进展。该项目的第二个组成部分将包括对系统的物理相关情况的调查，一个在流形上具有可变科里奥利参数的情况，以及对“对偶”变量中对应于奇异测度的解决方案的研究。自由边界问题出现在物理学、流体动力学、工程学和经济学的许多模型中。在物理系统中，“自由边界”是两个非常不同的状态之间条件快速变化的区域，例如气体动力学中的冲击波。在数学上，这种快速转变被简化为在控制物理的偏微分方程中沿着不连续的表面无限快速地发生。这个表面的位置一开始是不知道的，因此必须同时求解物理状态和它们的边界。近几十年来，自由边界问题的研究取得了重大进展。然而，在非线性偏微分方程的情况下，特别是在自由边界分隔的区域中具有非常不同性质的方程，仍然存在许多问题。主要研究人员计划应用自由边界问题技术研究气体动力学中一些基本的多维激波，特别是激波反射模式。这涉及到非线性方程和复杂结构系统的自由边界问题，需要新的方法来解决。了解自由边界的性质，例如它们的规律性、稳定性和几何形状，可以在模型和应用中更好地分析和使用数值方法。该项目的另一个重点是半匀转系统，该系统模拟大规模大气/海洋流动，并用于气象学，特别是大气锋面形成的模型。方法包括研究相关的monge - kantorovich型问题。Monge-Kantorovich质量传递理论最近在几个领域（例如，动力学理论，流体流动，弹性晶体，颗粒材料，城市规划，微观经济决策问题）得到了成功的应用。由于该项目解决了工程和气象方面的重要问题，因此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) 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.