The Ladyzhenskaya-Prodi-Serrin Conditions and the Search for Extreme Behavior in 3D Navier-Stokes Flows

Abstract

In this investigation, we conduct a systematic computational search for potential singularities in 3D Navier-Stokes flows on a periodic domain $\Omega$ based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that for a solution $\mathbf{u}(t)$ of the Navier-Stokes system to be regular on an interval $[0,T]$, the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$, where $2/p+3/q=1,\;q>3$, and the expression $\sup_{t \in [0,T]} \|\mathbf{u}(t)\|_{L^3}$ must be bounded. Flows which might become singular and violate these conditions are sought by solving a family of variational PDE optimization problems where we identify initial conditions $\mathbf{u}_{0}$ with the corresponding flows $\mathbf{u}(t)$ locally maximizing the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$ for a range of different values of $q$ and $p$ or the norm $\|\mathbf{u}(T)\|_{L^3}$ for different time windows $T$ and increasing sizes $\| \mathbf{u}_0 \|_{L^q}$ of the initial data. We consider two formulations where these expressions are maximized over appropriate Lebesgue spaces $L^q(\Omega)$ or the largest Hilbert-Sobolev spaces $H^s(\Omega)$ embedded in them. The lack of Hilbert-space structure in the first case necessitates development of a novel computational approach to solve the problem. While no evidence of unbounded growth of the quantities of interest, and hence also for singularity formation, was detected, we were able to quantify how "close" the flows realizing such worst-case scenarios come to forming a singularity. A comparison of these results with estimates on the rate of growth of the norms $||\mathbf{u}(t)||_{L^q}$ and of the enstrophy $\mathcal{E}(t)$ indicates that the extreme flows do enter a regime where these quantities are amplified at a rate consistent with singularity formation in finite time, but this growth is not sustained long enough for singularities to form.