The dynamics of equatorial long waves: a singular limit with fast variable coefficients
The dynamics of equatorial long waves: a singular limit with fast variable coefficients
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赤道长波动力学：快速变系数的奇异极限
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Majda
中科院分区：
文献类型：
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作者：
Alexandre Dutrifoy;A. Majda
在赤道，旋转产生的科里奥利力同样消失，因此赤道浅水方程的多时间尺度动力学自然会导致具有快速变系数的对称双曲系统的奇异极限。使用能量估计来获得更高空间导数的经典策略具有根本性的困难，因为形式上换向器项在极限中爆炸。这里，通过在涉及量子谐振子的升高和降低算子的合适的新变量中利用赤道浅水方程的特殊结构，并在基于埃尔米特算子的新函数空间中获得统一的高导数估计，可以避免这一基本困难。结果是一个全新的定理，描述了长波状态下赤道浅水方程的奇异极限，即使在一般不平衡的初始数据下，作为赤道长波方程的解。下面给出的结果指出了在其他物理相关奇异极限条件下对赤道浅水方程和赤道原始方程进行严格偏微分方程分析的方法。
At the equator, the Coriolis force from rotation vanishes identically so that multiple time scale dynamics for the equatorial shallow water equation naturally leads to singular limits of symmetric hyperbolic systems with fast variable coe-cients. The classical strategy of using energy estimates for higher spatial derivatives has a fundamental di-culty since formally the commutator terms explode in the limit. Here this fundamental di-culty is circumvented by exploiting the special structure of the equatorial shallow water equations in suitable new variables involving the raising and lowering operators for the quantum harmonic oscillator, and obtaining uniform higher derivative estimates in a new function space based on the Hermite operator. The result is a completely new theorem characterizing the singular limit of the equatorial shallow water equations in the long wave regime, even with general unbalanced initial data, as a solution of the equatorial long wave equation. The results presented below point the way for rigorous PDE analysis of both the equatorial shallow water equations and the equatorial primitive equations in other physically relevant singular limit regimes.