Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation
Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation
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DOI：
10.4310/cms.2016.v14.n7.a9
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发表时间：
2016
影响因子：
1
通讯作者：
Jiahong Wu;Xiaojing Xu;Liutang Xue;Z. Ye
中科院分区：
文献类型：
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作者：
Jiahong Wu;Xiaojing Xu;Liutang Xue;Z. Ye
不可压Boussinesq方程是物理学和Rayleigh-Bénard对流研究中的一个重要模型。一种推广是用分数拉普拉斯算子代替标准拉普拉斯算子，即速度方程中的（-Δ）α/2和温度方程中的（-Δ）β/2。本文研究了具有临界耗散（α+β=1）或超临界耗散（α+β<1）的二维不可压Boussinesq方程.我们证明了两个主要结果。第一个证明了临界Boussinesq方程经典解的整体时间存在性，其中α+β=1且0.7692 <$10 13 <α<1。第二部分证明了具有超临界耗散α+β<1和0.7692 × 1013 <α<1的Boussinesq方程的Leray-Hopf型弱解的最终正则性。
The incompressible Boussinesq equations serve as an important model in geophysics as well as in the study of Rayleigh–Bénard convection. One generalization is to replace the standard Laplacian operator by a fractional Laplacian operator, namely (−Δ)α/2 in the velocity equation and (−Δ)β/2 in the temperature equation. This paper is concerned with the two-dimensional (2D) incompressible Boussinesq equations with critical dissipation (α+β=1) or supercritical dissipation (α+β<1). We prove two main results. This first one establishes the global-in-time existence of classical solutions to the critical Boussinesq equations with α+β=1 and 0.7692≈ 10 13 <α<1. The second one proves the eventual regularity of Leray–Hopf type weak solutions to the Boussinesq equations with supercritical dissipation α+β<1 and 0.7692≈ 10 13 <α<1.