On the singularity formation of a 3D model for Incompressible Euler and Navier-Stokes equations

Monday, February 22, 2010 - 9:00am - 9:45am
EE/CS 3-180
Thomas Hou (California Institute of Technology)
Keywords: 3D incompressible Navier-Stokes equations,
finite time blow-up, and global regularity,
and stabilizing effect of convection.

Abstract: We study the singularity formation of a recntly proposed 3D model
for the incompressible Euler and Navier-Stokes equations. This
3D model is derived from the axisymmetric Navier-Stokes equations
with swirl using a set of new variables. The model preserves
almost all the properties of the full 3D Euler or Navier-Stokes
equations except for the convection term which is neglected. If we add the convection term back to our model, we would recover the
full Navier-Stokes equations. We will present numerical evidence
which supports that the 3D model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. Finally, we prove rigorously that the 3D model develops finite time singularities for a large class of initial data with finite energy and appropriate bounadry conditions. This work may shed interesting light into the stabilizing effect of convection for 3D incompressible Euler and Navier-Stokes equations.
MSC Code: