Finite Time Singularities of Solutions of the Euler Fluid Equations in 3D

Thursday, October 28, 2004 - 1:30pm - 2:20pm
EE/CS 3-180
Yves Pomeau (University of Arizona)
Joint work with D. Sciamarella.

The smoothness at all times of the solutions of the equations for inviscid incompressible fluids in 3D with finite energy remains a major unanswered question in applied mathematics. Long ago Leray suggested to look at the equations for self-similar singularities for viscous fluids, by assuming a point singularity in 3D. This way of approaching the question has been taken very rarely over the years. I will show, in the case of 3D Euler, that such a point singularity is very unlikely because of the conservation of energy and circulation, that puts very severe constraints on the possible singularities. Following this line of thought, one discovers that a singularity is possible along a line at a given time. Furthermore, a pure self similar collapse is also unlikely because it brings in a stationary flow with constant positive divergence in the collapsing frame. If nothing counteracts this positive divergence, anything is pulled to infinity in the long run and there is no non trivial solution of the self similar equations. Therefore one needs to have some sort of instability that brings features at smaller and smaller scales as one gets closer and closer to the singularity time. I'll sketch a possible scenario for such a quasi-self similar collapse in axissymetric geometry.