Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

TL;DR

This paper proves the mathematical validity of the Prandtl boundary layer theory for nonlinear nonhomogeneous incompressible Navier-Stokes equations in 3D plane parallel flows, demonstrating convergence of density and velocity under various norms.

Contribution

It generalizes the mathematical validation of the Prandtl boundary layer theory to nonhomogeneous flows, including convergence results in multiple Sobolev norms.

Findings

Convergence of density and velocity established.

Mathematical validation of boundary layer theory extended to nonhomogeneous flows.

Results include convergence in space-time uniform and $L^ abla(H^1)$ norms.

Abstract

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flow of nonhomogeneous incompressible Navier-Stokes equations. The convergence for the density and velocity are shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the $L^\infty(H^1)$ norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.