In this paper we deal with the system of the non-steady Navier-Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. Suppose that the system is solvable with some given data (the initial velocity and the right hand side). We prove that there exists a unique solution of this system for data which are small perturbations of the previous ones.

In this paper we consider the initial boundary value problem of the Navier-Stokes system with various types of boundary conditions. We study the global-in-time existence and uniqueness of a solution of this system. In particular, suppose that the problem is solvable with some given data (the initial velocity and the external body force). We prove that there exists a unique solution for data which are small perturbations of the previous ones.Mathematics Subject Classification. 35Q30, 35D05.

Abstract:In this paper we consider the system of the non-steady Navier-Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y , respectively, to be the space of "possible" solutions of this problem and the space of its data. We define the operator N : X → Y and formulate our problem in terms of operator equations. Let u ∈ X and G P u : X → Y be the Frechet derivative of N at u. We prove that G P u is one-to-one and onto Y . Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. Next result proved in the Appendix of this paper is W 2,2 -regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.

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 → Y P and formulate our problem in terms of operator equations. Let u ∈ X P and G P u : X P → Y P be the Frechet derivative of N P at u. Denote by M R the set of all functions u such that G P u is one-to-one and onto Y P . We prove that M R is weakly dense and weakly open.

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