Stability of Couette flow for 2D Boussinesq system with vertical dissipation
Stability of Couette flow for 2D Boussinesq system with vertical dissipation
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DOI：
10.1016/j.jfa.2021.109255
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发表时间：
2020-04
影响因子：
1.7
通讯作者：
Wen Deng;Jiahong Wu;Ping Zhang
中科院分区：
文献类型：
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作者：
Wen Deng;Jiahong Wu;Ping Zhang
本文建立了二维Boussinesq方程仅具有垂直耗散项的Couette流的非线性稳定性。这里涉及的Boussinesq方程模拟浮力驱动的流体，如大气和海洋流动。由于浮力的存在，标准Boussinesq方程的能量可以随时间增长。正是线性非自伴算子y∂x−ν∂y y在微扰方程中产生的增强耗散使非线性稳定性成为可能。当Couette流(y，0)的初始摄动不超过对适当幂的粘性时(在具有b>；4 3的Soblev空间Hb中)，我们证明了在T×R上只有垂直耗散的二维Boussinesq系统的解保持在同阶的Couette附近.这一结果的一个特殊结果是只有垂直耗散的二维Navier-Stokes方程Couette的稳定性。
This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator y∂ x− ν∂ y y in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow (y, 0) is no more than the viscosity to a suitable power (in the Sobolev space H b with b> 4 3), we prove that the solution of the 2D Boussinesq system with only vertical dissipation on T× R remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.