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

Regularity of Navier-Stokes flows is characterized by 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 $\mathbf{u}(t)$ is smooth on the interval $[0,T]$. Conversely, if this quantity is unbounded, a singularity must occur at some time $t \in [0,T]$. We consider 3D Navier-Stokes flows on a triply-periodic domain $\Omega$ and have probed these conditions for different values of $q$ by studying a family of variational optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$ 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 $\| \mathbf{u}(t) \|_{L^q(\Omega)}$. We also study inequalities involving the rate of growth of these norms which make it possible to assess how "close" these extreme flows come to producing a singularity.