Bounds on the enstrophy growth rate for solutions of the 3-d Navier-Stokes equations

It is still an open problem whether smooth solutions to the 3-d Navier-Stokes equations lose regularity in finite time. But it is known that if enstrophy ($\|\mathbf{\omega}\|^2$) remains finite, the solution is regular. The growth rate of enstrophy can be estimated from the Navier-Stokes equations by Sobelev inequalities. In general form, $d\|\mathbf{\omega}\|^2/dt \le c(\|\mathbf{\omega}\|^2)^\alpha$, where $c$ is a constant. In 2d, the exponent $\alpha$ is 2 and leads to regularity. However, $\alpha=3$ in 3d, which shows only finite-time regularity of the solutions. In these types of estimates, incompressibility is not used. We formulate the search for the maximal enstrophy growth rate as a variational problem and include incompessibility as a constraint. The variational problem is solved numerically by a gradient-flow type algorithm. Our preliminary results show that $\alpha\approx 1.75$, which hints that solutions of the 3-d Navier-Stokes equations are regular for all time.