Journal of the
Korean Mathematical Society JKMS

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in $H^1$ and the forcing function in $L^p([0, T); L^2)$, $T>0$, $2 \leq p \leq +\infty$ satisfy a certain condition. This condition commonly appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain $\epsilon$ tends to zero.

Keywords: Navier-Stokes equations, global existence, strong solution

MSC numbers: 35Q30, 35K51