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Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-05-29 , DOI: 10.1007/s00526-021-01976-w
Boqing Dong , Jiahong Wu , Xiaojing Xu , Ning Zhu

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system 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 \(\theta \) into the horizontal average \(({\bar{u}}, {\bar{\theta }})\) and the corresponding oscillation \(({\widetilde{u}}, {\widetilde{\theta }})\), 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 \(({\widetilde{u}}, {\widetilde{\theta }})\) decays exponentially to zero in \(H^1\) and \((u, \theta )\) converges to \(({\bar{u}}, {\bar{\theta }})\). This result reflects the stratification phenomenon of buoyancy-driven fluids.



中文翻译:

具有水平耗散的二维各向异性 Boussinesq 方程的稳定性和指数衰减

流体静力平衡是流体动力学和天体物理学中的一个突出话题。了解Boussinesq系统静水平衡附近的摄动稳定性有助于了解某些天气现象。此处关注的 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,因此很难解决稳定性和精确的长时间行为问题。当空间域为\({\ mathbb {R}} ^ 2 \)时,Sobolev设置中的稳定性问题仍然存在。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时间行为。通过分解速度u和温度\(\theta \)转化为水平平均值\(({\bar{u}}, {\bar{\theta }})\)和相应的振荡\(({\widetilde{u}}, {\widetilde{\theta }})\),并推导出各种各向异性不等式,我们能够在 Sobolev 空间\(H^2\) 中建立全局稳定性。此外,我们证明振荡\(({\widetilde{u}}, {\widetilde{\theta }})\)\(H^1\)\((u, \theta )\)收敛到\(({\ bar {u}},{\ bar {\ theta}})\)。该结果反映了浮力驱动流体的分层现象。

更新日期:2021-05-30
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