This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes equation --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no finite a priori bounds available for the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. To quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $\mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach. By solving these problems for a broad range of values of $T$ and $\mathcal{E}_0$, we demonstrate that the maximum growth of enstrophy appears finite and scales in proportion to $\mathcal{E}_0^{3/2}$. Thus, in the worst-case scenario the enstrophy remains bounded for all times and there is no evidence for formation of singularity in finite time. We will also review earlier results where a similar approach allowed us to probe the sharpness of a priori bounds on the growth of enstrophy and palinstrophy in 1D Burgers and 2D Navier-Stokes flows.
[Joint work with Dongfang Yun and Di Kang]
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