Boundary Layer Expansions for the Stationary Navier-Stokes Equations

TL;DR

This paper constructs an approximate solution to the stationary 2D Navier-Stokes equations using boundary layer theory, providing decay estimates and establishing global regularity of the Prandtl system without classical coordinate changes.

Contribution

It introduces a novel approach to prove global-in-x regularity of the Prandtl system without von-Mise coordinates, aiding in boundary layer stability analysis.

Findings

Constructed an approximate Navier-Stokes solution via Euler-Prandtl expansion.

Established sharp decay estimates for boundary layer quantities.

Proved global-in-x regularity of the Prandtl system independently of classical methods.

Abstract

This is the first part of a two paper sequence in which we prove the global-in-x stability of the classical Prandtl boundary layer for the 2D, stationary Navier-Stokes equations. In this part, we provide a construction of an approximate Navier-Stokes solution, obtained by a classical Euler- Prandtl asymptotic expansion. We develop here sharp decay estimates on these quantities. Of independent interest, we establish \textit{without} using the classical von-Mise change of coordinates, proofs of global in x regularity of the Prandtl system. The results of this paper are used in the second part of this sequence, [IM20] (arXiv:2008.12347), to prove the asymptotic stability of the boundary layer as $\eps \rightarrow 0$ and $x \rightarrow \infty$.