Solutions to the Navier-Stokes Equations with Mixed Boundary Conditions in Two-Dimensional Bounded Domains

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 $\mathcal{N}:X\rightarrow Y$ and formulate our problem in terms of operator equations. Let $\mathbf{u}\in X$ and ${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}: X\rightarrow Y$ be the Frechet derivative of $\mathcal{N}$ at $\mathbf{u}$. We prove that ${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}$ 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.