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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.

自由边界问题出现在物理学、工程学、流体动力学和经济学的许多模型中。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激波。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地发生。这个表面的位置是事先不知道的，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，对于非线性偏微分方程，特别是混合型偏微分方程，还有许多重要问题有待研究。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一些基本的多维激波，特别是激波反射模式。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这类问题。了解自由边界的性质，如规则性、稳定性和几何性质，可以在模型和应用中更好地分析和使用数值方法。该项目的另一个领域是半转地系统，这是一种以旋转为主的大气/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。PI计划继续研究半转矩模型中可变科里奥利参数的物理现实情况，并研究解的稳定性。该项目涉及工程和大气科学中的基础数学模型。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。本项目为研究生提供研究训练机会。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将继续对势流和全等熵欧拉系统的自相似激波反射进行研究。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和势流方程解的梯度中的不连续点。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，其中的未知数是椭圆区域和该区域的解。PI将继续研究正则反射全局解的存在性、稳定性和正则性，将全局解的存在性结果推广到可压缩欧拉系统和三维锥反射的情况。进一步的研究包括正则反射问题在各种解的稳定性。(2)利用Monge-Kantorovich质量输运方法研究半转矩方程组。PI将研究具有可变科里奥利参数的半转系统，这是一种考虑地球曲率的模型。PI还计划利用相对熵方法继续研究欧拉系统解到半转系统解的收敛性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。