Gravitational Effects on Rotating Stars and Deep Water Waves

Brown University, Providence RI

Investigators

Abstract

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.