论文标题
在有限通道中围绕COUETTE流动的二维Boussinesq系统的渐近稳定性
Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel
论文作者
论文摘要
在本文中，我们研究了在有限通道中的二维Navier-Stokes Boussinesq系统的渐近稳定性，其粘度$ν$和小的热扩散$μ$。特别是，我们证明，如果初始速度和初始温度$（v_ {in}，ρ_{in}）$满足$ \ | v_ {in} - （y，0）\ | _ { $ \ |ρ_{in} -1 \ | _ {h_x^{1} l_y^2} \ leq \ e_1 \ e_1 \ min \ min \ {c {c，μ\}^{\ f {\ f {\ f {11} {12} {12}}}} $ $ $ \ e_1 $ $ nim $ $ nim $ nim $ nim $ nim， Navier-Stokes boussinesq系统，速度保留在$ O（\ min \ {ν，μ\}^{\ f12}）$的$（\}^{\ f12}）$之内，并以$ t \ to t to \ infty $;温度保持在$ o（\ min \ {ν，μ\}^{\ f {\ f {11} {12}}）$的$ $ 1 $，并以$ 1 $为$ t \ to \ infty $。
In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $ν$ and small thermal diffusion $μ$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},ρ_{in})$ satisfies $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{ν,μ\}^{\f12}$ and $\|ρ_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{ν,μ\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $ν, μ$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min\{ν,μ\}^{\f12})$ of the Couette flow, and approaches to Couette flow as $t\to\infty$; the temperature remains within $O(\min\{ν,μ\}^{\f{11}{12}})$ of the constant $1$, and approaches to $1$ as $t\to\infty$.
