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

We derive the existence, uniqueness, and uniform $L^{p}$ 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 $L^{p}$ 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.