On vortex stretching and the global regularity of Euler flows I. Axisymmetric flow without swirl

The question of vortex growth in Euler flows leads naturally to the emergence of paired vortex structure and the “geometric” stretching of vortex lines. In the present paper, the first of two papers devoted to this question, we examine bounds on the growth of vorticity in axisymmetric flow without swirl. We show that the known bound on vorticity in this case, exponential in time, can be improved for large time by adhering closely to the geometric constraints imposed by the symmetry of the flow. and the conservation of the support of vorticity. Under appropriate conditions, the vorticity is shown to grow no faster that O(t). The kinematic vortex structure used to obtain this bound does not, however, conserve kinetic energy. If energy conservation is imposed, but not that of support volume, the bound is reduced to O(t). It appears that optimizing vorticity conserving both energy and volume will involve filamentary structures. We further propose that in the absence of the symmetry of the present class of flows, conservation of energy should be dropped from the local analysis of stretching of paired structures having variable stretching rates, and replaced by conservation of total energy, an idea which is explored further in the second paper.