Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

In this paper we are concerned with the stationary and non-stationary Stokes and Navier–Stokes problems with mixed boundary conditions involving velocity, pressure, rotation, stress and normal derivative of velocity together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we get variational formulations of the Stokes and Navier–Stokes problems with the mixed boundary conditions. Also, we study existence and uniqueness of solutions to the problems.