论文标题
2D各向异性Boussinesq方程的稳定性和指数衰减，并具有水平耗散
Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation
论文作者
论文摘要
静液压平衡是流体动力学和天体物理学中的重要主题。了解Boussinesq系统静水平衡附近的扰动的稳定性有助于深入了解某些天气现象。这里集中在这里的2D Boussinesq系统是各向异性的，仅涉及水平耗散和水平热扩散。由于缺乏垂直耗散，很难稳定和精确的大型行为问题。当空间域为$ \ mathbb r^2 $时，Sobolev设置中的稳定性问题保持打开状态。当空间域为$ \ Mathbb T \ Times \ Mathbb R $时，本文解决了稳定性问题，并指定了扰动的精确大型行为。通过将速度$ u $和温度$θ$分解为水平平均值$（\ bar u，\barθ）$以及相应的振荡$（\ widetilde u，\widetildeθ）$，并得出各种各向异性不平等现象，我们可以在sobole $ $ h^2 $中建立全球稳定性。此外，我们证明了振荡$（\ widetilde u，\widetildeθ）$衰减为$ h^1 $和$（u，θ）$在$ h^1 $中的零衰减，收敛到$（\ bar u，\barθ）$。该结果反映了浮力驱动的流体的分层现象。
The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial domain is $\mathbb T\times \mathbb R$, this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $θ$ into the horizontal average $(\bar u, \barθ)$ and the corresponding oscillation $(\widetilde u, \widetilde θ)$, and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space $H^2$. In addition, we prove that the oscillation $(\widetilde u, \widetilde θ)$ decays exponentially to zero in $H^1$ and $(u, θ)$ converges to $(\bar u, \barθ)$. This result reflects the stratification phenomenon of buoyancy-driven fluids.
