Partial Differential Equations Seminar

Stars are commonly modelled as a spherically symmetric gas under the influence of either Newtonian or relativistic gravitational forces, giving rise to either the Euler-Poisson or Einstein-Euler equations. In certain regimes, it is known that smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In the relativistic setting, in certain regimes, the blow-up creates a so-called naked singularity. In this talk, I will present the nonlinear stability of a particular family of these blow-up profiles, the (relativistic) Larson-Penston solution. This is based on joint works with Yan Guo, Mahir Hadzic, and Juhi Jang.

We discuss Aleksandrov estimates for kinetic equations in non-divergence form with a rough diffusion coefficient. We present two different mechanisms, each of which has its advantages.