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

  • Wendong Wang

    Dalian University of Technology, China
  • Di Wu

    South China University of Technology, Guangzhou, China
  • Zhifei Zhang

    Peking University, Beijing, China
Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations cover
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Abstract

The classical Ladyzhenskaya–Prodi–Serrin regularity criterion states that if the Leray weak solution uu of the Navier–Stokes equations satisfies uLq(0,T;Lp(R3))u\in L^q(0,T; L^p(\mathbb{R}^3)) with 2q+3p1\frac{2}{q} +\frac{3}{p} \leq 1, p>3p>3, then it is regular in R3×(0,T)\mathbb{R}^3\times (0,T). In this paper, we prove that the Leray weak solution is also regular in R3×(0,T)\mathbb{R}^3\times (0,T) under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., u3Lq,1(0,T;Lp(R3))u_3\in L^{q,1}(0,T; L^p(\mathbb{R}^3)) with 2q+3p1\frac{2}{q} +\frac{3}{p} \leq 1, 3<p<+3<p<+\infty. 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