Small scale formation in ideal fluids

The incompressible Euler equation of fluid mechanics describes motion of ideal fluid, and goes back to 1755. In two dimensions, global regularity of solutions is known, and double exponential in time upper bound on growth of the derivatives of solution goes back to 1930s. I will describe a construction of example showing sharpness of this upper bound, based on work joint with Vladimir Sverak. The construction has been motivated by a singularity formation scenario proposed by Hou and Luo for the 3D Euler equation. If time permits, I will also describe some recent work on model equations designed to gain insight into possible singularity formation in solutions of 3D Euler equation, which remains a major open problem.