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.