Nonlinear free boundary and evolution problems

非线性自由边界和演化问题

• 批准号: 1401490
• 负责人: Mikhail Feldman
• 金额: $ 21.36万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401490&HistoricalAwards=false
• 关键词: Nonlinear; free; boundary; evolution; 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. 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.

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。更好地理解自由边界的性质，如稳定性、规律性和几何结构，使得在模型和应用中研究复杂现象成为可能。本项目的第一个领域是研究气体动力学中的冲击反射问题，这是最基本的多维冲击波问题之一。本研究涉及非线性偏微分方程自由边界问题分析中的几个具有挑战性的问题。项目的另一个领域是半转地系统，一个以旋转为主的大气/海洋流动模型。它具有丰富的基于蒙格-坎托洛维奇质量传递理论的数学结构。我们计划研究半转矩模型中可变科里奥利参数的物理现实情况。由于该项目涉及工程和大气科学中的基本数学模型，因此将产生更广泛的影响。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。研究生将参与该项目。该项目包括两个主要主题：(1)冲击分析中的自由边界问题。PI将继续他的工作，自相似激波反射的势流和全等熵欧拉系统。在许多物理环境中都会出现冲击反射问题。此外，这些问题在多维守恒定律的数学理论中是重要的，因为它们的解是可压缩流体多维欧拉方程通解的基石和渐近吸引子。可压缩流体动力学的自相似方程是混合椭圆-双曲型方程。激波对应于欧拉方程组解中的不连续点和势流方程解的梯度。在整个激波中，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述为一个自由边界问题，其中未知的是椭圆（亚音速）区域和椭圆区域的解。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，这是气体动力学的基本模型。进一步研究了正则反射问题的唯一性和稳定性。(2)拟研究的另一个领域是半营养系统。PI将研究黎曼流形上具有可变科里奥利参数的半转系统。这种模型是由于考虑了地球的曲率而产生的。PI还计划研究低规律性的解决方案，这来自于对具有中性稳定区域的流动的建模。这些项目包括研究Monge-Kantorovich质量输运问题，以及具有低规则性矢量场的输运方程的拉格朗日解。