Regularity for solutions to the Navier-Stokes equations with one velocity component regular. (English) Zbl 0993.35072

He, Cheng

Electron. J. Differ. Equ. 2002, Paper No. 29, 13 p. (2002).

Summary: We establish a regularity criterion for solutions to the Navier-Stokes equations, which is only related to one component of the velocity field. Let \((u, p)\) be a weak solution to the Navier-Stokes equations. We show that if any component of the velocity field \(u\), for example \(u_3\), satisfies either \(u_3 \in L^\infty(\mathbb{R}^3\times (0, T))\) or \(\nabla u_3 \in L^p (0, T; L^q(\mathbb{R}^3))\) with \(1/p + 3/2q = 1/2\) and \(q \geq 3\) for some \(T > 0\), then \(u\) is regular on \([0, T]\).

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D10	Regularity of generalized solutions of PDE (MSC2000)

Keywords:

Navier-Stokes equations; weak solutions; regularity