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

Shijin Ding; Zhilin Lin; Dongjuan Niu

doi:10.3934/dcds.2020193

Article Contents

2020, Volume 40, Issue 8: 4579-4596. Doi: 10.3934/dcds.2020193

This issue Previous Article Next Article Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise

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

1.
South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3.
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

^* Corresponding author: Zhilin Lin
^* Corresponding author: Zhilin Lin

Received: November 2018

Revised: October 2019

Published online: August 2020

Ding's research is supported by the National Natural Science Foundation of China (No.11371152, No.11571117, No.11871005 and No.11771155) and Guangdong Provincial Natural Science Foundation (No.2017A030313003). Lin's research is supported by the Innovation Project of Graduate School of South China Normal University (No.2018LKXM009). Niu's research is supported by the National Natural Science Foundation of China (No.11471220 and No.11871046) while she was visiting at the Institute of Mathematical Sciences of the Chinese University of Hong Kong.

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Abstract

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is 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.

Keywords:

Boundary layer,
plane parallel channel flows,
nonhomogeneous incompressible Navier-Stokes equations,
Prandtl theory.

Mathematics Subject Classification: 35Q30, 76N10.

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Figure 1. The plane parallel channel flow in $ Q = [0, L]^2 \times [0, 1] $.

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