Stability Threshold of the Couette Flow for Navier–Stokes Boussinesq System with Large Richardson Number \(\boldsymbol{\gamma}^{\boldsymbol{2}} \boldsymbol{\gt} \frac{\boldsymbol 1}{\boldsymbol 4}\)

Abstract.

In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely, the Richardson number \(\gamma^2\gt \frac{1}{4}\) . Precisely, we prove that if the initial perturbation \((u_{in},\vartheta _{in})\) of the Couette flow \(v_s=(y,0)\) and the linear temperature \(\rho _s=-\gamma^2y+1\) satisfies \(\|u_{in}\|_{H^{s+1}}+\|\vartheta _{in}\|_{H^{s+2}}\leq \epsilon _0\nu ^{\frac 12}\) , then the asymptotic stability holds.

Keywords

  1. Navier–Stokes Boussinesq
  2. Couette flow
  3. large Richardson number
  4. stability threshold

MSC codes

  1. 76D05
  2. 35Q30
  3. 76E05

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Acknowledgment.

C. Zhai appreciates the hospitality of NYU.

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 1284 - 1318
ISSN (online): 1095-7154

History

Submitted: 10 May 2022
Accepted: 4 October 2022
Published online: 27 April 2023

Keywords

  1. Navier–Stokes Boussinesq
  2. Couette flow
  3. large Richardson number
  4. stability threshold

MSC codes

  1. 76D05
  2. 35Q30
  3. 76E05

Authors

Affiliations

Cuili Zhai
School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, People’s Republic of China.
Weiren Zhao Contact the author
Department of Mathematics, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates.

Funding Information

Funding: The work of the first author was supported by a grant from the China Scholarship Council while visiting the center SITE, NYU Abu Dhabi, and was also partially supported by the National Natural Science Foundation of China grant 12201035.

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