The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions

Petr Kučera

doi:10.3934/dcdss.2010.3.325

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2010, Volume 3, Issue 2: 325-337. Doi: 10.3934/dcdss.2010.3.325

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The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions

Petr Kučera^1,

1.
Czech Technical University, Faculty of Civil Engineering, Department of Mathematics, Thakurova 7, Prague 16629

Received: June 2009

Revised: October 2009

Published: June 2010

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Abstract

In this paper we deal with the system of periodic Navier-Stokes equations with mixed boundary conditions. We define Banach spaces X_P and Y_P , respectively, the space of "possible'' solutions of this problem and the space of its data. We define the operator N_P : X_p $\to$ Y_P and formulate our problem in terms of operator equations. Let u $\in$ X_P and g_{P u} : X_P $\to$ Y_P be the Frechet derivative of N_P at u . Denote by M_R the set of all functions u such that g_Pu is one-to-one and onto Y_P . We prove that M_R is weakly dense and weakly open.

Keywords:

Navier-Stokes equations,
mixed boundary conditions.

Mathematics Subject Classification: Primary: 35Q30, 35J65; Secondary: 76D03, 76D05.

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