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 N:X→Y and formulate our problem in terms of operator equations. Let u∈X and GPu:X→Y be the Fréchet derivative of N at u. We prove that GPu 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. The next result proved in the Appendix of this paper is W2, 2‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.