Georgia Institute of Technology College of Sciences

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifold of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows. This is a joint work with Zhiwu Lin and Chongchun Zeng.

TBA by Xilu Zhu

Astronomy is arguably the oldest scientific discipline. Precise measurements of the motion of celestial bodies date back to the ancient Babylonians, Chinese, Greeks, and indigenous peoples outside Eurasia. Starting in the 19th century, systematic applications of physical principles to the formation and dynamics of stars marked the birth of astrophysics as a subfield of physics. Present-day astrophysics employs an array of theoretical and observational tools to construct sophisticated and predictive models of the origin, evolution, and death of stars.

While stars can be largely described within Newtonian physics, some of their most interesting properties, such as bounds on their mass-radius ratio, their potential collapse into a black hole, or effects of viscosity on gravitational waves emitted by mergers of neutron stars, can only be studied via applications of general relativity. Moreover, as a matter of principle, we ought to be able to fully understand stars as general-relativistic phenomena. The mathematical treatment of stars within general relativity, however, has lagged behind. Little progress has been made on this front since the discovery of the Tolman-Oppenheimer-Volkoff (TOV) equations and the Oppenheimer-Snyder solution in the late 1930s. The TOV equations describe a static (i.e., time independent), perfectly spherically symmetric star, whilst the latter describes the collapse of a perfectly spherically symmetric star with no pressure into a black hole. Despite being landmark results in general relativity, both situations are highly idealized. Inferences about generic properties of general-relativistic stars derived from such models are, therefore, a priori unjustified.

In this talk, I will discuss the problem of formulating a sound mathematical theory of general-relativistic star evolution based on the Einstein-Euler system. After setting up the problem, I will explain its main challenges, but also discuss the rich physics and mathematics involved in its study. A fundamental difficulty involves understanding the mathematics of the fluid-vacuum interface which separates the body of the star from vacuum. This interface displays singular behavior which is not amenable to current mathematical techniques. This difficulty, however, can be circumvented if we consider stars that are spherically symmetric but not static. The resulting evolution problem corresponds to a dynamic (i.e., time-dependent) generalization of the TOV equations.

This is joint work with Jared Speck.

On September 8, OpenAI announced a Lean-certified resolution of the Navier-Stokes Millennium Problem by furnishing a classical solution to the forced 3d Navier-Stokes equations which blows up at the origin in finite time. The purpose of this talk will be to provide (i) historical context and relevance for the Millennium Problem; and (ii) give an overall view of the techniques developed towards its resolution from the last ten years or so. I will avoid technical details, and endeavor to make the talk accessible to those outside PDE. While I will briefly address the ongoing priority dispute and allegations of misconduct by OpenAI, this will not be the focus of the talk.