Title: Regularity and blow up in ideal fluid
Abstract: The Euler equation of fluid mechanics describes motion of inviscid and incompressible fluid, and has been derived in 1755. In two dimensions, smooth initial data lead to globally regular solutions. The upper bound on the growth of derivatives of solutions is double exponential in time, and goes back to works of Wolibner and Holder in 1930s. In three dimensions, the question of global regularity vs finite time blow up remains open.
I will discuss a construction of a solution to the 2D Euler equation in a disk where the gradient of vorticity does grow at a double exponential rate. This is based on work joint with Sverak. This example is inspired by numerical simulations by Hou and Luo, who propose a new scenario for finite time blow up in the 3D Euler equation. If time permits, I will mention some results on simplified models designed to gain insight into the three dimensional setting.