# American Institute of Mathematical Sciences

June  2010, 3(2): 325-337. doi: 10.3934/dcdss.2010.3.325

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

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

Received  June 2009 Revised  October 2009 Published  April 2010

In this paper we deal with the system of periodic Navier-Stokes equations with mixed boundary conditions. We define Banach spaces XP and YP , respectively, the space of "possible'' solutions of this problem and the space of its data. We define the operator NP : Xp $\to$ YP and formulate our problem in terms of operator equations. Let u $\in$ XP and gP u : XP $\to$ YP be the Frechet derivative of NP at u . Denote by MR the set of all functions u such that gPu is one-to-one and onto YP . We prove that MR is weakly dense and weakly open.
Citation: Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325
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