Collapse of a class of three-dimensional euler vortices

Substitution of the flow field U(x,y,z,t) = {u(x,y,t),v(x,y,t),z/y(x,y,t}} into the three-dimensional incompressible Euler equations generates a closed system of evolution equations, for the strain rate ?(0;, y,t) and the two-dimensional vorticity ui(x, 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 j(x,y,t) appears to develop a finite-time singularity, from smooth initial data, in the region where 7 is negative. Here, we prove that, for a large class of initial data, the support of ?^- := max{0, -?} necessarily collapses to zero in a finite time, while at the same time, the L1 norm of ? remains non-zero. Hence, ? 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. © 2000 The Royal Society.