The incompressible Euler equation of fluid mechanics has been derived in
1755. It is one of the central equations of applied analysis, yet due to
its nonlinearity and locality many fundamental properties of the solutions
remain poorly understood. In particular, the global regularity vs finite
time blow up question for 3D Euler equation remains open.
In two dimensions, it has been known since 1930s that solutions to Euler
equation with smooth initial data are globally regular. The best upper
bound on the size of derivatives of the solution is double exponential in
time. I will describe a construction showing that such generation of small
scales can actually happen, so that the double exponential bound is
qualitatively sharp. This is based on a joint work with Vladimir Sverak.
Our work has been motivated by numerical experiments due to Hou and Luo who
propose a new scenario for singularity formation in solutions of 3D Euler
equation. The scenario is axi-symmetric. The geometry of the scenario is
related to the geometry of our 2D Euler example and involves hyperbolic
points of the flow located on the boundary of the domain. I will discuss
two one-dimensional models that have been developed to gain insight into
the behavior of the solutions in the 3D setting.