The classical Ladyzhenskaya–Prodi–Serrin regularity criterion states that if the Leray weak solution of the Navier–Stokes equations satisfies with , , then it is regular in . In this paper, we prove that the Leray weak solution is also regular in under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., with , . This result means that if the solution blows up at a time, then all three components of the velocity have to blow up simultaneously.
Cite this article
Wendong Wang, Di Wu, Zhifei Zhang, Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2023),DOI 10.4171/AIHPC/77