On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation

We show a new lower bound on the $\dot{H}^{\frac{3}{2}}\left(\mathbb {T}^3\right)$Ḣ32T3 norm of a possible blow-up solution to the Navier-Stokes equation, and also comment on the extension of this result to the whole space. This estimate can be seen as a natural limiting result for Leray's blow-up estimates in $L^p\left(\mathbb {R}^3\right)$LpR3, 3 < p < ∞. We also show a lower bound on the blow-up rate of a possible blow-up solution of the Navier-Stokes equation in $\dot{H}^{\frac{5}{2}}\left(\mathbb {T}^3\right)$Ḣ52T3, and give the corresponding extension to the case of the whole space.