Long time existence of regular solutions to non-homogeneousNavier-Stokes equations

Wojciech M. Zajączkowski

doi:10.3934/dcdss.2013.6.1427

Article Contents

2013, Volume 6, Issue 5: 1427-1455. Doi: 10.3934/dcdss.2013.6.1427

This issue Previous Article Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions Next Article

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

Wojciech M. Zajączkowski^1,

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw

Received: November 30, 2011

Revised: March 31, 2012

Published: September 2013

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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.

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Mathematics Subject Classification: Primary: 35A01, 35Q30; Secondary: 76D05, 76D03, 35Q35.

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