Global Steady Prandtl Expansion over a Moving Boundary II

TL;DR: In this article, it was shown that Navier-Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid limit globally in the presence of a sufficiently small velocity mismatch.

Abstract: This is the second paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, $$y = 0$$, can be decomposed into Euler and Prandtl flows in the inviscid limit globally in $$[1, \infty ) \times [0,\infty )$$, assuming a sufficiently small velocity mismatch. In this paper, we develop a functional framework to capture precise decay rates of the remainders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential derivatives. These tools are then used in conjunction with a third-order energy analysis, which, in particular, enables us to control the nonlinearity $$vu_y$$ globally, leading to the main a priori estimate in the analysis.