Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound u _L∞ₜL3,∞ₓ≤ M ‖ u ‖ L t ∞ L x 3 , ∞ ≤ M . Namely, we show that if T^* T ∗ is a first blow-up time and (0,T^*) ( 0 , T ∗ ) is a singular point then aligned u(· ,t) _L³(B₀(R))≥ C(M)log (1/T^*-t ),\,\,\,\,\,\,R=O((T^*-t)1/2-). aligned ‖ u ( · , t ) ‖ L 3 ( B 0 ( R ) ) ≥ C ( M ) log ( 1 T ∗ - t ) , R = O ( ( T ∗ - t ) 1 2 - ) . We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on R³× (0,1) R 3 × ( 0 , 1 ) with aligned ₙ u(· ,t₍ₙ₎) _L³(R³)<∞ \,\,\,and\,\,\,t₍ₙ₎↑ 1, aligned sup n ‖ u ( · , t ( n