Bounds on the enstrophy growth rate for solutions of the three-dimensional Navier-Stokes equations.

Abstract

It is still an open problem whether smooth solutions to the 3D Navier-Stokes equations lose regularity in finite time. But it is known that if the enstrophy ( w</fen></fen> 22 ) remains finite, the solution is regular. The growth rate of the enstrophy can be estimated from the Navier-Stokes equations by Sobolev inequalities. In general form, dw</fen></fen> 22/dt&le;c w</fen></fen>2 2</fen>a , where <italic>c</italic> is a constant. In 2D the exponent alpha is 2 and leads to regularity. However, alpha = 3 in 3D, which yields only finite-time regularity of the solutions. In these estimates, incompressibility is not used. We formulate the maximal enstrophy growth rate as a variational problem and include incompressibility as a constraint. The variational problem is solved numerically by a gradient-flow type algorithm. Our results show that alpha = 1.78 for small enstrophies and alpha = 3 as enstrophy gets larger. Thus the Sobolev bounds are actually realizable even with incompressibility constraint.