Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations

Abstract

The classical Ladyzhenskaya-Prodi-Serrin regularity criterion states that if the Leray weak solution u of the Navier-Stokes equations satisfies u ϵ Lq(0, T; LP(ℝ3)) with [Formula presented] p > 3, then it is regular in ℝ3 × (0,T). In this paper, we prove that the Leray weak solution is also regular in ℝ3 × (0, T) under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., u3 ϵ Lq,1 (0, T; LP(ℝ3)) with [Formula presented] ≤ 1,3 < p < + ∞. This result means that if the solution blows up at a time, then all three components of the velocity have to blow up simultaneously.