Navier-Stokes Problems with Small Parameters in Half-Space and Application

We derive the existence, uniqueness, and uniform LP estimates for the abstract Navier- Stokes problem with small parameters in half-space. The equation involves small parameters and an abstract operator in a Banach space E. Hence, we obtain the singular perturbation property for the Stokes operator depending on a parameter. We can obtain the various classes of Navier-Stokes equations by choosing E and the linear operators A. These classes occur in a wide variety of physical systems. As application we establish the existence, uniqueness, and uniform LP estimates for the solution of the mixed problems for infinitely many Navier-Stokes equations and nonlocal mixed problems for the high order Navier-Stokes equations.