准地转运动稳定性的数学方法

Mathematical methods of stability for quasigeostrophic motions

在准地转运动的非线性稳定性的研究中,先验不等式起着至关重要的作用.所得到的不等式越精细,得到的非线性稳定性的结论就越好.本文主要分两个部分,第一部分综合性地介绍了变分原理以及利用变分原理及分析的技巧得到一系列重要的最佳不等式的方法.这些不等式中有些是已有的结果,只是用更直接的方法重新得到,还有一些小等式是首次发现的.第二部分以准地转运动的非线性稳定性的研究为主线,介绍了有关的理论;利用第一部分的数学基础理论对已有的准地转运动的非线性稳定性定理作了统一的处理,使得在论证思路上更为清晰,在证明方法上更为简洁.

The a priori estmates play an important role in the study of nonlinear stability of quasigeostophic motions. The tighter the estimate, the better the nonlinear stability criterion is. There are two main parts in this paper. In the first part, the variational principle was introduced and a series of best possible inequalities were derived by the principle and analysis, in which some known results were recovered by more direct and simpler mothods, and some of which were newly obtained. In the second part, relevent theory of quasigeostophic motions was introduced and some known results of nonlinear stability of quasigeostophic motions were recovered uniformly with simpler and clearer reduction and treatment.