Vortex dynamics and the existence of solutions to the Navier–Stokes equations

The Biot–Savart model for a vortex filament predicts a finite time singularity in which the maximum velocity diverges as (t*−t)−1/2 for the time t tending to t*. The filament pairs with itself, yet remains locally smooth even though the characteristic length scales as (t*−t)1/2. A multiscale perturbative treatment of the Euler equations is developed for solutions that are locally a two-dimensional vortex dipole centered on a slowly varying three-dimensional space curve. For short periods of time the Euler and Biot–Savart solutions agree. Provided this correspondence persists, a sufficiently small viscosity ν will not control the divergence in the maximum velocity until it is of order exp(cst/ν), where cst is a constant of order the filament circulation. Singularities in the Navier–Stokes equations cannot be easily dismissed. The most questionable step in the arguments presented occurs for ν=0, namely whether the Euler vortex dipole solutions break down when they self-stretch.