Enstrophy, Enstrophy Production, and Regularity in the 3D Navier-Stokes Equations

It is still not known whether solutions to the 3D Navier-Stokes equations for incompressible flows in a finite periodic box can become singular in finite time. (This is the subject of one of the $1M Clay Prize problems.) It is known that a solution remains smooth as long as the enstrophy, i.e., the mean-square vorticity, of the solution is finite. The generation rate of enstrophy is given by a functional that can be bounded using elementary functional estimates. Those estimates establish short-time regularity but they do not rule out finite-time singularities. In the original research reported here we formulate and solve a variational problem for the maximal growth rate of enstrophy and display flows that generate enstrophy at the greatest possible rate. Implications of the results for questions of regularity or singularity of the 3D Navier-Stokes equations are discussed. This is joint work with Lu Lu (Wachovia Investments).

Charles R. Doering, University of Michigan, Ann Arbor

 

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