论文标题
2D Boussinesq系统的COUETTE流量的稳定性，垂直耗散
Stability of Couette flow for 2D Boussinesq system with vertical dissipation
论文作者
论文摘要
本文建立了仅垂直耗散的2D Boussinesq方程的COUETTE流的非线性稳定性。这里涉及的BoussinesQ方程模型浮力驱动的流体，例如大气和海洋流动。由于存在浮力强迫，标准Boussinesq方程的能量可能会及时增长。这是由线性非自动 - 附加操作员$ y \ y \ partial_x-ν\ partial_ {yy} $在扰动方程中创建的增强耗散，这使得非线性稳定性成为可能。当Couette Flow $（Y，0）$的初始扰动不超过适当功率的粘度（在Sobolev Space $ H^B $带有$ b> \ frac43 $的$ H^B $）时，我们证明了2D BoussnesQ系统的解决方案仅在$ \ \ \ Mathbb t Times \ MathBB $ close close collow to collows collows collows collows collows cout to collows cout to collow of ticles close。该结果的一个特殊结果是COUETTE对于仅垂直耗散的2D Navier-Stokes方程的稳定性。
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 $y\partial_x -ν\partial_{yy}$ 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 $b>\frac43$), we prove that the solution of the 2D Boussnesq system with only vertical dissipation on $\mathbb T\times \mathbb R$ 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.
