Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and stratified temperature

Wenting Huang, Zekai Luo, Ying Sun, Xiaojing Xu

Abstract

This paper examines the stability threshold at high Reynolds numbers $\textbf{Re}$ for the three-dimensional Boussinesq equations with rotation on the domain $Ω=\{(x,\,y,\,z)\in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\}$ around the Couette flow $(y,0,0)$ and the vertically stratified temperature $Θ_s=1+α^2 z$. For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy $\left\|u_{\mathrm{in}}\right\|_{H^{N+2}\cap W^{N+3,1}}+\left\|θ_{\mathrm{in}}\right\|_{H^{N+1}\cap W^{N+3,1}}<δ\mathbf{Re}^{-\frac{14}{15}}$, for any $N\geq 11$ and some $δ>0$ independent of $\mathbf{Re}$, then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.

Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and stratified temperature

Abstract

This paper examines the stability threshold at high Reynolds numbers for the three-dimensional Boussinesq equations with rotation on the domain around the Couette flow and the vertically stratified temperature . For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy , for any and some independent of , then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.
Paper Structure (30 sections, 23 theorems, 315 equations)

This paper contains 30 sections, 23 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Known results for the Navier-Stokes equation with/without rotation
  4. Motivation behind \ref{['1.1']} with vertical stratification
  5. Summary of main result
  6. Obstacles and strategies
  7. Linear stability
  8. Difficulty 1:
  9. Strategy 1: Constructing good unknowns.
  10. Difficulty 2:
  11. Strategy 2: Constructing the dispersion combined quantities
  12. Nonlinear stability
  13. Difficulty 1: Structural issue for zero modes
  14. Strategy 1: Quasilinear method for zero modes.
  15. Difficulty 2: Regularity issue for non-zero modes
  16. ...and 15 more sections

Key Result

Theorem 1.1

Assume that $\nu>0$, $\alpha>0$, $\beta \in \mathbb{R}$ with $B_{\beta}>0$. Let $u_{\mathrm{in}}$ with a divergence-free, smooth vector field, then the solutions $u=u_{\neq}+\widetilde{u}_0+\overline{u}_0$ and $\theta=\theta_{\neq}+\widetilde{\theta}_0+\overline{\theta}_0$ to the linearized equation $(2)$ Cancellation of lift-up effect in the Bradshow-Richardson stable regime: For $k=0$, if $l \ne

Theorems & Definitions (27)

  • Theorem 1.1: Linear stability
  • Remark 1.2
  • Theorem 1.3: Nonlinear stability
  • Remark 1.4
  • Lemma 2.1: Van der Corput lemma SM1993
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Corollary 3.4
  • Lemma 3.5
  • ...and 17 more