Navier-Stokes problems with small parameters in half space and application

Abstract

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.