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Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel

Qionglei Chen, Zhen Li

Abstract

Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on Bénard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow $(U(t,y),0)$ with a constant background temperature $a\in\mathbb{R}$. The analysis is performed in the finite channel $\mathbb{T}\times[0,1]$ with non-slip boundary condition. By means of the sharp resolvent estimates and space-time estimates, we establish that the Boussinesq system admits a globally stable solution around the monotone shear flow, provided that the initial perturbation satisfies $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}, \|\langle D_x\rangle θ^{\mathrm{in}}\|_{L^2} \leq cν^{\frac56}$. Moreover, we derive the enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity.

Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel

Abstract

Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on Bénard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow with a constant background temperature . The analysis is performed in the finite channel with non-slip boundary condition. By means of the sharp resolvent estimates and space-time estimates, we establish that the Boussinesq system admits a globally stable solution around the monotone shear flow, provided that the initial perturbation satisfies . Moreover, we derive the enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity.
Paper Structure (4 sections, 7 theorems, 108 equations)

This paper contains 4 sections, 7 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Space-Time estimates of the linearized Boussinesq equations
  3. Nonlinear stability
  4. Auxiliary estimates

Key Result

Theorem 1.1

Let $(u,\theta)$ be the solution to pertu. There exist positive constants $\nu_0$, $\epsilon_{0}$, $c$, such that for $0<\nu\leq \nu_{0}$, if the initial perturbation satisfies then the solution $(u,\theta)$ satisfies the global stability estimates where the stability norms are given by with $f_k(t,y)=:\int_{\mathbb{T}} f(t,x,y) e^{-ikx}dx$ and $\|\cdot\|_{L^pL^q}=\|\cdot\|_{L^p_tL^q_y}$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['Th: tran thre']}
  • Lemma A.1
  • ...and 2 more