Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability threshold for 2D shear flows of the Boussinesq system near Couette

Dongfen Bian, Xueke Pu

Abstract

In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain $\mathbb{T} \times \mathbb{R}$. The main goal is to prove the nonlinear stability of the shear flow $(U^S,Θ^S)=((e^{νt\partial_{yy}}U(y),0)^{\top},αy)$ with $U(y)$ close to $y$ and $α\geq0$. We separate two cases: one is $α\geq 0$ small scaling with the viscosity coefficients and the case without smallness of $α$ and fixed heat diffusion coefficient. The novelty here is that we don't require $μ=ν$ and only need to assume that $μ$ is scaled with $ν$ or fixed, where $μ$ is the inverse of the Reynolds number and $ν$ is the heat diffusion coefficient.

Stability threshold for 2D shear flows of the Boussinesq system near Couette

Abstract

In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain . The main goal is to prove the nonlinear stability of the shear flow with close to and . We separate two cases: one is small scaling with the viscosity coefficients and the case without smallness of and fixed heat diffusion coefficient. The novelty here is that we don't require and only need to assume that is scaled with or fixed, where is the inverse of the Reynolds number and is the heat diffusion coefficient.

Paper Structure

This paper contains 5 sections, 4 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Stability
  3. Change of dependent variables
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Theorem \ref{['thm2']}

Key Result

Theorem \oldthetheorem

Let $N\geq5$, $\nu,\mu\in(0,1)$ with $\nu\lesssim \mu$, and for some small $\delta$ independent of $\nu$ and $\mu$. There exists $\gamma_1$ and $\gamma_2$ such that if and $0\leq \alpha<\nu^{1/3}\mu^{1/2}\varepsilon_2/ \varepsilon_1$, then the solution with initial data $(\omega_{in},\theta_{in})$ is global in time and for any $T>0$, it holds and where the implicit constants do not depend on $

Theorems & Definitions (7)

  • Theorem \oldthetheorem: $\alpha\ll1$
  • Remark 1.1
  • Theorem \oldthetheorem: $\alpha$ not small
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2