Collapse of a class of three-dimensional Euler vortices

Substitution of the flow field U(x, y, z, t) = {u(x, y, t), upsilon (x, y, t), z gamma (x, y, t)} into the three-dimensional incompressible Euler equations generates a closed system of evolution equations, for the strain rate y(z,y,t) and the two-dimensional vorticity omega (z, y, t), which are uniform in the z-direction. The system models a class of dynamical, stretched three-dimensional vortex flows that include Burgers' vortices. Recent numerical simulations by Ohkitani & Gibbon have revealed that the strain rate gamma (z, y, t) appears to develop a finite-time singularity, from smooth initial data, in the region where y is negative. Here, we prove that, for a large class of initial data, the support of gamma (-) := max{0, -gamma} necessarily collapses to zero in a finite time, while at the same time, the L-1 norm of gamma (-) remains non-zero. Hence, gamma (-) must necessarily become singular before or at the time of collapse. Our vortex flow represents one of a subclass of Euler solutions that have infinite energy. The fundamental question of finite-time singularity formation from smooth initial data for finite-energy three-dimensional Euler solutions remains the important open question.