Gravitational Effects on Rotating Stars and Deep Water Waves

这个项目的目的是调查在流体流动研究中出现的几个重要的数学问题，这些流体流动受到重力的显著影响。这些问题的两个主要来源是旋转恒星和星系的模型，以及海洋中深水波的模型。研究自转恒星的平衡形状和密度分布是数学物理史上的一个经典问题。早期的努力始于18世纪，致力于寻找密度恒定的椭球体形状。在二十世纪，通过求解描述可压缩流体的方程取得了重大进展，从而使得具有可变气体密度的恒星的模型更加逼真。构造这种解的数学方法包括将相同数量的气体放入不同形状和密度分布的气体中，并找到物理能量最少的气体。然而，这种方法只提供了解存在的证据，没有提供太多关于恒星形状的信息，也没有提供太多关于恒星自转速度变化时解如何变化的信息。在这个项目中，研究人员的目标是使用一种不同的方法提供新的存在证明，该方法包括连续变形一颗不旋转的圆形恒星的形状，以得出旋转的恒星解。新方法旨在返回有关形状和对转速的依赖关系的信息。对于深水波，研究人员将研究某些重要的模型方程，这些方程描述了海洋表面以下的内波以及大气中的壮观现象，例如在澳大利亚东北部看到的“牵牛花”云。这些模型所描述的内波具有实际意义；例如，内波可以与海面波相互作用，产生可损坏或摧毁海洋船只或固定结构的无赖波。更具体地说，该项目旨在利用隐函数定理型摄动技术证明Euler-Poisson和Vlasov-Poisson方程的旋转星解的存在性。除了证明涉及适用于该问题的空间的详细分析估计外，研究者还需要通过证明关于核中函数的某一积分-微分方程零的定理来研究线性化算子。该研究将利用拓扑度参数研究解曲线向更快转速的延拓，并发现解曲线可能的破裂机制。此外，研究人员计划使用变分方法和谱分析来研究Euler-Poisson方程的旋转白矮星解的质量界限现象。对于涉及深水波的模型，该项目将探索本杰明-小野方程的完全可积方面。在前人工作的基础上，研究者将研究在Fokas-Ablowitz逆散射变换中提出的Jost解的存在性、唯一性和渐近性质。这些问题与导数算子某些奇异积分摄动的连续谱密切相关。

This project aims to investigate several important mathematical questions arising in the study of fluid flows that are significantly influenced by gravity. Two major sources of these questions are models of rotating stars and galaxies, and models of deep water waves in the ocean. The study of equilibrium shape and density distribution of rotating stars is a classical question in the history of mathematical physics. Early efforts beginning in the eighteenth century were devoted to finding ellipsoidal shapes with constant density. In the twentieth century, major progress was made by solving equations describing compressible fluids, thus allowing a more realistic model of stars with variable gaseous density. The mathematical method used to construct such solutions involves putting the same amount of gas in different shapes and density distributions, and finding one with the least physical energy. However, such a method only provides proof for existence of solutions, without delivering too much information on the shape of the star, or how the solution varies as the rotation speed of the star varies. In this project, the investigator aims to provide new existence proofs using a different method, which involves continuously deforming the shape of a round, non-rotating star to arrive at a rotating star solution. The new method is intended to return information about shape and dependence on rotation speed. For deep water waves, the investigator will study certain important model equations that describe internal waves below the ocean surface as well as spectacular phenomena in the atmosphere, such as the "Morning Glory" cloud seen in northeastern Australia. The internal waves described by such models have practical significance; for example, internal waves can interact with surface ocean waves to produce rogue waves that can damage or destroy ocean vessels or fixed structures.More specifically, the project aims to prove existence of rotating star solutions to the Euler-Poisson and Vlasov-Poisson equations, using an implicit function theorem type perturbation technique. Apart from proving detailed analytical estimates involving spaces adapted to the problem, the investigator will need to study the linearized operator by proving a vanishing theorem about a certain integro-differential equation for functions in the kernel. The research will study continuation of the solution curve to faster rotation speed using topological degree arguments, and discover possible breakdown mechanisms of the solution curve. In addition, the investigator plans to examine the phenomenon of mass bound on rotating white dwarf solutions to the Euler-Poisson equations, using both variational methods and spectral analysis. For models involving deep water waves, the project will explore the completely integrable aspect of the Benjamin-Ono equation. Building on previous work, the investigator will study existence, uniqueness, and asymptotic properties of the Jost solutions proposed in the Fokas-Ablowitz inverse scattering transform. These questions are closely related to the continuous spectrum of certain singular integral perturbations of the derivative operator.