Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

Abstract

We consider the motion of incompressible viscous non-homogene-ous fluid described by the Navier-Stokes equations in a bounded cylinder $\Omega$ under boundary slip conditions. Assume that the $x_3$-axis is the axis of the cylinder. Let $\varrho$ be the density of the fluid, $v$ -- the velocity and $f$ the external force field. Assuming that quantities $\nabla\varrho(0)$, $\partial_{x_3}v(0)$, $\partial_{x_3}f$, $f_3|_{\partial\Omega}$ are sufficiently small in some norms we prove large time regular solutions such that $v\in H^{2+s,1+s/2}(\Omega\times(0,T))$, $\nabla p\in H^{s,s/2}(\Omega\times(0,T))$, $½ < s < 1$ without any restriction on the existence time $T$. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.