Global Steady Prandtl Expansion over a Moving Boundary II

Abstract

In this three-part monograph, 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, ∞) × [0, ∞), assuming a sufficiently small velocity mismatch. Sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers. We then 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.

Separation for the stationary Prandtl equation

“…Let us also mention recent works by Guo and Nguyen [10] and by Iyer [13,12,14], in which the authors justify the Prandtl expansion either over a moving plate or over a rotating disk. Note that in these two cases, the velocity of the boundary layer on the boundary is non zero, which somehow prevents recirculation and separation.…”

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

“…The desired estimate (12) for U + ν k ,θ (c k ) can be easily deduced from the estimate of U + θ,ν k (ĉ k ). We prove the estimates in (12) for {U + ν k ,θ }, the proof for {U − ν k ,θ } is similar.…”

“…The desired estimate (12) for U + ν k ,θ (c k ) can be easily deduced from the estimate of U + θ,ν k (ĉ k ). We prove the estimates in (12) for {U + ν k ,θ }, the proof for {U − ν k ,θ } is similar. In the following, C denotes various constant depending only on c. By Lemma 2.2, ||U + ν k ,θ || L ∞ (−1,1) ≤ C for all k. By Theorem A, the convergence of {c k } to c and the fact that min [−1,1] P c > 0 and c 1 , c 2 > 0, we have, for large k, To prove the second estimate in (12), we first prove the following lemma.…”

“…There have been a large amount of research work on vanishing viscosity limit for incompressible Navier-Stokes equations. See for instance [1], [2], [4], [5], [6], [7], [10], [11], [12], [20], [21], [26], [27], [33] and [36].…”

“…Then for any ǫ > 0 and any positive integer m, there exists some constant C, depending only on ǫ, m andĉ, such that for large k, (12), the ǫ could not be taken as 0 in general.…”