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 non-locality many fundamental properties of the solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional 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 available upper bound on the size of derivatives of the solution has been double exponential in time.
I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp.
This 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 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.