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

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

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