Abstract

This investigation concerns a systematic computational search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy $\mathcal{E}(t)$ 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. Another well-known conditional regularity result are the Ladyzhenskaya-Prodi-Serrin conditions asserting that the quantity $\mathcal{L}\int_0^{T \| \mathbf{u}(t) \|}{Lq(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, must remain bounded if the solution $\mathbf{u}(t)$ is smooth on the interval $[0,T]$. However, there are no finite a priori bounds available for these quantities and hence the regularity problem for the 3D Navier-Stokes system remains open. To quantify the maximum possible growth of $\mathcal{E}(T)$ and $\mathcal{L}$, we consider families of PDE optimization problems in which initial conditions are sought subject to certain constraints so that these quantities in the resulting Navier-Stokes flows are maximized. These problems are solved computationally using a large-scale adjoint-based gradient approach. By solving these problems for a broad range of parameter values we demonstrate that the maximum growth of $\mathcal{E}(T)$ and $\mathcal{L}{q,p}$ appears finite and follows well-defined power-law relations in terms of the size of the initial data. Thus, in the worst-case scenarios the two quantities remain bounded for all times and there is no evidence for singularity formation 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.