Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound

First, we show that if the pressure p (associated to a weak Leray– Hopf solution v of the Navier–Stokes equations) satisfies(Formula Presented) then v possesses higher integrability up to the first potential blow-up time T∗. Our method is concise and is based upon energy estimates applied to powers of |v| and the utilization of a “small exponent”. As a consequence, we show that if a weak Leray–Hopf solution v first blows up at T∗ and satisfies the Type I condition ‖v‖L∞ t (0,T∗;L3,∞ (R3)) ≤ M, then (Formula Presented) This is the first result of its kind, improving the integrability exponent of ∇v under the Type I assumption in the three-dimensional setting. Finally, we show that if v: R3 ×[−1, 0] → R3 is a weak Leray–Hopf solution to the Navier–Stokes equations with sn ↑ 0 such that (Formula Presented) then v possesses at most O(M20) singular points at t = 0. Our method is direct and concise. It is based upon known ε-regularity, global bounds on a Navier– Stokes solution with initial data in L3,∞ (R3) and rescaling arguments. We do not require arguments based on backward uniqueness nor unique continuation results for parabolic operators