Gravitational Effects on Rotating Stars and Deep Water Waves

该项目旨在研究在受重力显著影响的流体流动研究中出现的几个重要数学问题。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。旋转恒星的平衡形状和密度分布的研究是数学物理史上的经典问题。从世纪开始的早期努力致力于寻找具有恒定密度的椭球形状。在世纪，通过求解描述可压缩流体的方程取得了重大进展，从而允许具有可变气体密度的恒星的更现实的模型。用于构建此类解决方案的数学方法包括将相同数量的气体置于不同的形状和密度分布中，并找到具有最小物理能量的气体。然而，这种方法只提供了解的存在性的证明，而没有提供太多关于星星形状的信息，或者解如何随着星星的旋转速度变化而变化。在这个项目中，研究人员的目标是使用不同的方法提供新的存在性证明，该方法包括连续变形圆形非旋转星星的形状，以获得旋转星星的解决方案。新方法旨在返回有关形状和旋转速度依赖性的信息。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。该模型所描述的内波具有实际意义，例如，内波可以与表面海浪相互作用，产生可以损坏或摧毁海洋船只或固定结构的异常波。具体而言，该项目旨在使用隐函数定理型摄动技术证明Euler-Poisson和Vlasov-Poisson方程的旋转星星解的存在性。除了证明详细的分析估计，涉及空间适应的问题，调查人员将需要研究线性化运营商证明消失定理的某个积分微分方程的功能在内核。本研究将使用拓扑度参数研究解曲线向更快转速的延续，并发现解曲线可能的破裂机制。此外，研究人员计划研究的现象，质量约束旋转白色矮解决方案的欧拉-泊松方程，使用变分法和频谱分析。对于涉及深水波的模型，该项目将探讨Benjamin-Ono方程的完全可积性。在以前工作的基础上，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的某些奇异积分扰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.