Existence and Stability Analysis for Nonlinear Free Boundary and Evolution Problems

在物理、工程、流体力学和经济学的许多模型中都会出现自由边界问题。自由边界是条件在两个非常不同的状态之间快速变化的区域，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沿着支配物理的偏微分方程式中的不连续表面发生无限快。这个表面的位置事先是未知的，因此必须求解物理状态和它们的边界。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。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此类问题。了解自由边界的性质，如正则性、稳定性和几何性质，可以在模型和应用中更好地分析和数值方法。该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。PI计划继续研究半地转模式中变Coriolis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。该项目为研究生提供了研究培训机会。该项目包括两个主要主题：(1)激波分析中的自由边界问题。PI将继续研究势流和全等熵欧拉系统的自相似激波反射。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间断和势流方程解的梯度的间断。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以椭圆区域及其解为未知量的自由边界问题。PI将继续研究正则反射整体解的存在性、稳定性和正则性，将整体存在性结果推广到可压缩欧拉系统和三维锥体反射的情形。进一步的研究包括正则反射问题在各类解中的稳定性。(2)用Monge-Kantorovich质量输运方法研究半地转方程组。PI将研究具有可变科里奥利参数的半地转系统，这是一个考虑地球曲率的模型。PI还计划继续研究Euler系统的解与半地转系统的解的收敛，使用相对熵方法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.