Existence and Stability Analysis for Nonlinear Free Boundary and Evolution Problems

非线性自由边界和演化问题的存在性和稳定性分析

基本信息

批准号：
2054689
负责人：
Mikhail Feldman
金额：
$ 27.4万
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：
2021
资助国家：
美国
起止时间：
2021-06-15 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054689&HistoricalAwards=false
关键词：
Existence Stability Analysis Nonlinear Free

项目摘要

Free boundary problems arise in many models in physics, engineering, fluid dynamics, and economics. Free boundaries are regions of rapid variations of conditions between two very different states, such as shock waves 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. The location of this surface is not known in advance, thus one must solve both for physical states and their boundaries. Significant progress in the study of free boundary problems has been made during the last several decades. However, in the case of nonlinear partial differential equations, and especially equations of mixed type, many important questions are yet to be studied. The principal investigator (PI) 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 having a complex structure, and thus new methods need to be developed to handle such problems. Understanding properties of free boundaries, such as regularity, stability and geometric properties, allows for a better analysis and numerical methods in models and applications. Another area of the project is the semigeostrophic system, a model of rotation-dominated atmospheric/ocean flows. It exhibits a rich mathematical structure based on Monge-Kantorovich mass transport theory. The PI plans to continue the study of the physically realistic case of variable Coriolis parameter in the semigeostrophic model, and also study stability properties of solutions. The project addresses fundamental mathematical models in engineering and atmospheric sciences. Closer interaction with the engineering and meteorological communities is one of the priorities of the project. The project provides research training opportunities for graduate students. The project consists of two main topics: (1) Free boundary problems in shock analysis. The PI will continue work on self-similar shock reflection for potential flow and for the 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 to the Euler system and in the gradient of the solution for potential flow equation. The type of the equation may change from hyperbolic to elliptic across the shock. The shock reflection problem can be formulated as a free boundary problem in which the unknowns are the elliptic region and the solution in that region. The PI will continue work on the existence, stability, and regularity of global solutions to the regular reflection, to extend the global existence results to the case of compressible Euler system and three-dimensional reflection by a cone. Further study includes stability for the regular reflection problem in various classes of solutions. (2) The study of the system of semigeostrophic equations, using methods from Monge-Kantorovich mass transport. The PI will study the semigeostrophic system with variable Coriolis parameter, which is a model that arises from taking into account the curvature of the Earth. The PI also plans to continue the study of convergence of solutions of the Euler system to solutions of the semigeostrophic system using relative entropy methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在物理，工程，流体动力学和经济学的许多模型中都出现了自由边界问题。自由边界是两个非常不同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。从数学上讲，这种快速的转变被简化为沿着偏微分方程的不连续性表面无限快速发生，该方程是对物理学的。该表面的位置未提前知道，因此必须解决物理状态及其边界。在过去的几十年中，在研究自由边界问题的研究中取得了重大进展。但是，在非线性部分微分方程，尤其是混合类型的方程式的情况下，许多重要的问题尚待研究。首席研究员（PI）计划应用自由边界问题的技术来研究气体动力学中的一些基本多维冲击波，特别是冲击反射模式。这涉及到具有复杂结构的非线性方程和系统的自由边界问题，因此需要开发新的方法来处理此类问题。了解自由边界的属性，例如规律性，稳定性和几何特性，可以在模型和应用中进行更好的分析和数值方法。该项目的另一个领域是半神经化系统，这是一种旋转主导的大气/海洋流量的模型。它表现出基于Monge-Kantorovich质量运输理论的丰富数学结构。 PI计划继续研究半神经化模型中可变科里奥利参数的物理现实案例，并研究溶液的稳定性。该项目涉及工程和大气科学中的基本数学模型。与工程和气象社区的紧密互动是该项目的优先事项之一。该项目为研究生提供了研究培训机会。该项目由两个主要主题组成：（1）冲击分析中的自由边界问题。 PI将继续进行潜在流动和全等欧拉系统的自相似冲击反射。在许多物理情况下出现了冲击反射问题。此外，此类问题在多维保护定律的数学理论中很重要，因为它们的解决方案是针对可压缩流体的多维欧拉方程的一般解决方案的基础和渐近吸引者。可压缩流体动力学的自相似方程是混合的椭圆形纤维类型的。冲击对应于欧拉系统解决方案中的不连续性以及溶液的梯度以进行电势流程方程。在冲击中，方程的类型可能从双曲线变为椭圆形。冲击反射问题可以作为一个自由边界问题，其中未知数是椭圆形区域和该区域中的解决方案。 PI将继续致力于定期反思的全球解决方案的存在，稳定性和规律性，以将全球存在的结果扩展到可压缩的Euler系统和三维反射的情况下。进一步的研究包括在各种解决方案中定期反思问题的稳定性。（2）使用Monge-Kantorovich质量传输的方法研究半神经方程的系统。 PI将使用可变的科里奥利参数研究半神经系统，该模型是由考虑地球的曲率而产生的。 PI还计划继续使用相对熵方法研究Euler系统的解决方案对半神经系统解决方案的收敛性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来进行评估的。