On Serrin interior regularity criterion for Navier-Stokes equations

We revisit Serrin's interior spatial regularity criterion for distributional solutions to the Navier-Stokes equations in ℝ 3 and considerably relax the hypotheses in two main directions. More precisely, if 𝑢 ∈ 𝐿 𝑠 ′ 𝑡 ⁢ 𝐿 𝑠 𝑥 locally is a distributional solution to the Navier-Stokes equations with 2 𝑠 ′ + 3 𝑠 = 1 for 𝑠 ′ ∈ [ 4 , ∞ ) , then 𝑢 ∈ 𝐿 𝑞 𝑡 ⁡ ( 𝐶 ∞ 𝑥 ) locally for all 𝑞 ∈ ( 2 , 𝑠 ′ ) . If 𝑠 ′ ∈ ( 2 , 4 ) , the same conclusion holds provided that in addition 𝑢 ∈ 𝐿 4 𝑡 ⁡ ( 𝐿 𝑝 𝑥 ) locally, for some 𝑝 > 1 . In particular, we remove any integrability hypothesis on the vorticity, and we reduce the requirement of integrability in time all the way to 𝐿 4 from 𝐿 ∞ . To achieve this, we employ a new bootstrap argument, distinct from Serrin's, and we argue that a reduction of the exponent in time integrability does not follow from Serrin's original argument.