# Seminars & Events for Analysis Seminar

##### Regularity and blow up in models of fluid mechanics

I will discuss a family of modified SQG equations that varies between 2D Euler and SQG with patch-like initial data defined on half-plane. The family is modulated by a parameter that sets the degree of the kernel in the Biot-Savart law. The main result I would like to describe is the phase transition in the behavior of solutions that happens right beyond the 2D Euler case. Namely, for the 2D Euler equation the patch solution stays globally regular, while for a range of nearby models there exist regular initial data that lead to finite time blow up. The geometry of the blow up example involves hyperbolic fixed point of the flow at the boundary. If time permits, I will discuss some other recent works designed to better understand vorticity growth in a similar setting for solutions of the 3D Euler equation.

##### Dynamical relativistic liquid bodies

In this talk, I will discuss a new approach to establishing the well-posedness of the relativistic Euler equations for liquid bodies in vacuum. The approach is based on a wave formulation of the relativistic Euler equations that consists of a system of non-linear wave equations in divergence form together with a combination of acoustic and Dirichlet boundary conditions. The equations and boundary conditions of the wave formulation differs from the standard one by terms proportional to certain constraints, and one of the main technical problems to overcome is to show that these constraints propagate, which is necessary to ensure that solutions of the wave formulation determine solutions to the Euler equations with vacuum boundary conditions.