The interplay between analysis and computation in the study of 3D Euler singularities.

Tom Hou (Caltech)

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. A potential singularity in the 3D Euler equations is significant because it may be responsible for the onset of energy cascade in turbulent flows. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder. A careful local analysis also suggests that the blowing-up solution is highly anisotropic and is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. We have recently proved rigorously that the 1D model develops finite time singularity. Finally, we present some recent progress in developing an integrated analysis and computation strategy to analyze the finite time singularity of the original 3D Euler equations.