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PreprintarXiv licencearXiv

Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel

  • Jinkai Li
  • Zhilin Lin
  • Dong Wang
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Abstract

In this paper, we study the hydrodynamics stability of the 3-D plane Couette flow vs=(y,0,0) with constant temperature state ρs for the Boussinesq system in a finite channel 𝕋× [βˆ’1,1]Γ— 𝕋 with non-slip boundary condition in y direction. We prove that if the initial velocity field vin and initial temperature ρin satisfy |vinβˆ’(y,0,0)|H2 ≀ c0 Ξ½, |ρinβˆ’Οs|H2 ≀ c0 Ξ½2, for some c0>0 independent of the viscosity and thermal diffusion Ξ½, then the solutions to 3D Boussinesq system do not transit from the Couette flow and constant temperature state. To our best knowledge, this is the first result about the nonlinear stability of plane Couette flow for 3-D Boussinesq system in the domain with non-slip boundary condition. This result, together with [18], implies that the strong boundary layer effect does not lead to strong instability.

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Cite

BibTeXRIS
BibTeX
@misc{li2026nonlinear,
  title = {{Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel}},
  author = {Jinkai Li and Zhilin Lin and Dong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36989},
  eprint = {2609.36989},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36989},
}
RIS
TY  - UNPB
TI  - Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel
AU  - Jinkai Li
AU  - Zhilin Lin
AU  - Dong Wang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36989
UR  - https://arxiv.org/abs/2609.36989
AB  - In this paper, we study the hydrodynamics stability of the 3-D plane Couette flow v_s=(y,0,0) with constant temperature state ρ_s for the Boussinesq system in a finite channel 𝕋× [βˆ’1,1]Γ— 𝕋 with non-slip boundary condition in y direction. We prove that if the initial velocity field v_in and initial temperature ρ_in satisfy |v_inβˆ’(y,0,0)|_H^2 ≀ c_0 Ξ½, |ρ_inβˆ’Ο_s|_H^2 ≀ c_0 Ξ½^2, for some c_0>0 independent of the viscosity and thermal diffusion Ξ½, then the solutions to 3D Boussinesq system do not transit from the Couette flow and constant temperature state. To our best knowledge, this is the first result about the nonlinear stability of plane Couette flow for 3-D Boussinesq system in the domain with non-slip boundary condition. This result, together with [18], implies that the strong boundary layer effect does not lead to strong instability.
ER  -