Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations
Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations
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DOI：
10.1007/s00205-011-0411-5
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发表时间：
2010-10
影响因子：
2.5
通讯作者：
D. Chae;P. Constantin;Jiahong Wu
中科院分区：
文献类型：
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作者：
D. Chae;P. Constantin;Jiahong Wu
二维不可压缩欧拉方程的任何经典解在时间上都是全局的。然而，地表准地转方程的经典解是否始终保持其正则性仍然是一个悬而未决的问题。本文研究了一族主动标量方程的解，其中速度场的每个分量由标量θ通过确定，其中是Riesz变换，Λ =（-Δ）1/2。二维欧拉涡度方程对应于特殊情况P（Λ）= I，SQG方程对应于特殊情况P（Λ）= Λ。我们开发了工具来定义一类算子P，并建立了Loglog-Euler方程的全局正则性，其中P（Λ）=（log（I+ log（I-Δ）γ，0 <$γ <$1.此外，还得到了P（Λ）= Λβ且0 <$β <$1的模型的正则性准则.
Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each componentujof the velocity fielduis determined by the scalarθthrough, whereis a Riesz transform and Λ = (−Δ)1/2. The two-dimensional Euler vorticity equation corresponds to the special caseP(Λ) =Iwhile the SQG equation corresponds to the caseP(Λ) = Λ. We develop tools to boundfor a general class of operatorsPand establish the global regularity for the Loglog-Euler equation for whichP(Λ) = (log(I+ log(I− Δ)))γwith 0 ≦γ≦ 1. In addition, a regularity criterion for the model corresponding toP(Λ) = Λβwith 0 ≦β≦ 1 is also obtained.