Nonlinear Free Boundary and Evolution Problems

自由边界问题出现在物理学、工程学、流体动力学和经济学中的许多模型中。自由边界对应于描述问题的变量的急剧变化。虽然自由边界问题的研究已经取得了显著的进展，但对于非线性偏微分方程，特别是混合型方程，还有许多重要的问题有待研究。对于这些情况，更好地理解自由边界的性质，如稳定性，规则性和几何结构，将使人们有可能研究在现实世界中的应用复杂的现象。该项目的主要重点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。该项目还将通过与工程和气象界的密切互动以及通过对研究生的培训产生广泛的影响。该项目包括两个主要专题：（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完整的等熵欧拉系统。激波反射问题在许多物理情况下都会出现。此外，这些问题是重要的多维守恒律的数学理论，因为它们的解决方案是积木和渐近吸引子的一般解决方案的多维欧拉方程的可压缩流体。可压缩流体动力学的自相似方程是椭圆-双曲混合型的。激波对应于欧拉方程解的不连续性和位流方程解的梯度不连续性。方程的类型可以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，其中未知数是椭圆（亚音速）区域和椭圆区域内的解。PI将继续研究正则反射的整体解的存在性、稳定性和正则性，将整体解的存在性结果推广到可压缩Euler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在不同解类下的唯一性和稳定性。(2)拟研究的另一个领域是半地转系统。 PI将研究具有可变科里奥利参数的半地转系统。该奖项反映了NSF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。

Free boundary problems arise in many models in physics, engineering, fluid dynamics, and economics. Free boundaries correspond to sharp changes in the variables describing the problem. While significant progress has been made in the study of free boundary problems, many important questions have yet to be studied in the case of nonlinear partial differential equations and especially equations of mixed type. For these cases, a better understanding of properties of free boundaries, such as stability, regularity and geometric structure, would make it possible to study complex phenomena in real-world applications. The primary focus of this project is the study of shock reflection problems in gas dynamics, one of the most fundamental multidimensional shock wave problems. The project will also have a broad impact through close interactions with engineering and meteorological communities and through training of graduate students.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 in various classes of solutions. (2) Another area of the proposed research is semi-geostrophic system. The PI will study semi-geostrophic system with variable Coriolis parameter. Such model arises from taking into account the curvature of the Earth.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.