Small scale creation in inviscid fluids
Alexander Kiselev (Rice
The problem of global regularity vs
finite time blow up for the classical 3D equations of fluid mechanics is one
of the well known open problems of applied
analysis. Recently, a new scenario for finite time blow up of solutions to 3D
Euler equation has been proposed by Hou and Luo
based on extensive numerical simulation.
We will describe some recent
analytic developments stimulated by Hou and Luo
work. We will begin with the two-dimensional Euler equation, where an example
leading to the double exponential growth for all times in the vorticity
gradient has been constructed. We will then discuss some models that help
bridge the gap between two and three dimensions, and may help to gain insight
into the nature of possible blow up in 3D Euler.