Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.