Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $ε\in (1,2]$

Abstract

For a solution u to the Navier-Stokes equations in spatial dimension n ≥ 3 which blows up at a finite time T > 0, we prove the blowup estimate u(t) Ḃsp+ǫ, where sp := −1 + n p is the scaling-critical regularity, and ϕ is the cutoff function used to define the Littlewood-Paley projections. For ǫ = 2, we prove the same type of estimate but only for q = 1:Under the additional restriction that p, q ∈ [1, 2] and n = 3, these blowup estimates are implied by those first proved by Robinson,