On a variational inequality for the Navier-Stokes operator with variable viscosity

G. M. de Araújo; S. B. de Menezes

doi:10.3934/cpaa.2006.5.583

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2006, Volume 5, Issue 3: 583-596. Doi: 10.3934/cpaa.2006.5.583

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On a variational inequality for the Navier-Stokes operator with variable viscosity

G. M. de Araújo^1, and
S. B. de Menezes^1,

1.
Universidade Federal do Pará, Departamento de Matematica-CCEN, 66.075-110 Belém Pará

Received: May 31, 2005

Revised: January 31, 2006

Published: August 2006

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Abstract

In this paper we investigate the unilateral problem for the operator $L$ perturbed of Navier-Stokes operator in a cylindrical case, where

$Lu=u'-(\nu_0+\nu_1||u(t)||^2)\Delta u+(u.\nabla )u-f+\nabla p.$

The mixed problem for the operator $L$ was proposed by J. L. Lions [6]. Using an appropriate penalization, we obtain a variational inequality for the Navier-Stokes perturbed system.

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Mathematics Subject Classification: Primary: 35K60, 35Q30; Secondary: 35F30.

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