Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations
Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations
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广义大尺度半地转方程的全局适定性
DOI：
10.1007/s00205-012-0587-3
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发表时间：
2013
影响因子：
2.5
通讯作者：
S. Vasylkevych
中科院分区：
文献类型：
--
作者：
M. Çalık;M. Oliver;S. Vasylkevych
本文证明了具有周期边界条件的广义大规模半地转方程整体经典解的存在唯一性。这类快速旋转浅水的哈密顿平衡模型包括R. Salmon在1985年和2006年由第二作者概括。在初始位涡为正的物理约束下，所得结果与二维理想流体流动的欧拉方程的结果一样强。此外，我们确定了一个特殊的情况下，在索伯列夫空间中的速度场是两个导数平滑相比，一般情况下。我们的研究结果是基于仔细的估计表明，虽然位涡反演是非线性的，边界上的位涡反演算子的导数位涡保持线性。这使得适应的参数的基础上ellipticLP理论，提出了由Yudovich在1963年证明存在性和唯一性的弱解的二维欧拉方程，我们特定的非线性情况。
We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes theL1model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on ellipticLptheory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.
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