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

We show a new lower bound on the Ḣ3/2(T³)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 Lᵖ(R³)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 Ḣ5/2(T³)Ḣ52T3, and give the corresponding extension to the case of the whole space.