The Ladyzhenskaya-Prodi-Serrin Conditions and the Navier-Stokes Blow-Up Problem

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. One of the most important conditional regularity results are 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 this quantity is unbounded, a singularity must occur at some time $t \in [0,T]$. We have probed this condition by studying a family of variational optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$ for different $T$ and subject to certain constraints. Such problems are solved computationally using a large-scale adjoint-based gradient approach. Even in this worst-case scenario, no evidence has been found for singularity formation which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^q(\Omega)}$. However, the maximum enstrophy in these 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.