Nonlinear free boundary and evolution problems

自由边界问题自然而然地出现在物理学、流体力学、经济学的许多模型中。自由边界对应于描述问题的变量的急剧变化。在过去的几十年里，自由边界问题的研究取得了重大进展。然而，对于非线性偏微分方程组，特别是混合型方程，许多重要的问题仍有待研究。更好地理解自由边界的性质，如稳定性、正则性和几何结构，使研究模型和应用中的复杂现象成为可能。本项目的第一个领域是研究气体动力学中的激波反射问题，这是最基本的多维激波问题之一。这项研究涉及到分析非线性偏微分方程组自由边界问题的几个挑战性困难。项目的另一个领域是半地转系统，这是一种以旋转为主的大气/海洋流动模型。它展示了基于Monge-Kantorovich质量传输理论的丰富的数学结构。我们计划研究半地转模式中变Coriolis参数的物理现实情况。该项目将产生更广泛的影响，因为该项目涉及工程和大气科学中的基本数学模型。与工程界和气象界更密切的互动是该项目的优先事项之一。研究生将参与这个项目。项目包括两个主要主题：(1)冲击分析中的自由边界问题。PI将继续他在势流和全等熵欧拉系统的自相似激波反射方面的工作。震动反射问题在许多物理情况下都会出现。此外，这些问题在多维守恒律的数学理论中是重要的，因为它们的解是可压缩流体的多维欧拉方程通解的积木块和渐近吸引子。可压缩流体动力学的自相似方程是椭圆-双曲型混合方程。激波对应于欧拉系统解的不连续性和势流方程解的梯度不连续性。在激波中，方程的类型可能从双曲型变为椭圆型。激波反射问题可以表示为一个自由边界问题，其中未知数为椭圆(亚音速)区域和椭圆区域内的解。PI将继续他关于正则反射整体解的存在性、稳定性和正则性的工作，将整体存在性结果推广到可压缩欧拉系统的情况，这是气体动力学的一个基本模型。进一步的研究包括正则反射问题的唯一性和稳定性。(2)提出的另一个研究领域是半地转系统。PI将研究黎曼流形上具有可变科里奥利参数的半地转系统。这种模型是在考虑了地球的曲率后产生的。PI还计划具有低规律性的研究解决方案，这些解决方案来自于对具有中性稳定区域的流动进行建模。这些项目涉及Monge-Kantorovich质量输运问题的研究，以及具有低正则向量场的输运方程的拉格朗日解。

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.