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