On the Serrin-type condition on one velocity component for the Navier-Stokes equations

Abstract

In this paper we consider the regularity problem of the Navier-Stokes equations in ${\mathbb{R}}^3$. We show that the Serrin-type condition imposed on one component of the velocity $u_3\in L^p(0,T;L^q({\mathbb{R}}^3))$ satisfying $\frac{2}{p}+\frac{3}{q}<1$, $3<q\le +\infty$ implies the regularity of the weak Leray solution $u:{\mathbb{R}}^3\times (0,T)\to {\mathbb{R}}^3$ with the initial data belonging to $L^2({\mathbb{R}}^3)\cap L^3({\mathbb{R}}^3)$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

arXiv comment25 pages

References in corpus (1)

• Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

Cited by in corpus (4)

• Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations
• An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range
• Quantitative transfer of regularity of the incompressible Navier-Stokes equations from ${\mathbb{R}}^3$ to the case of a bounded domain
• Prodi--Serrin condition for 3D Navier--Stokes equations via one directional derivative of velocity