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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >On a Description of Long-time Behaviour of Dissipative Perturbations of Infinite Dimensional Hamiltonian Systems
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On a Description of Long-time Behaviour of Dissipative Perturbations of Infinite Dimensional Hamiltonian Systems

机译：关于无穷维哈密顿系统的耗散摄动的长时间行为的描述

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摘要

We present some recent results on long-time behaviour and limit regimes for a class of nonlinear partial differential equations which are dissipative perturbations of Hamiltonian systems. Contrasting with unperturbed case there exist finite dimensional global attractors for the considered equations. In a rather unified framework we construct infinite families of approximate inertial manifolds (AIMs) which are finite dimensional smooth surfaces in a phase space of the system whose small vicinities attract all solutions and contain the global attractor. Using the properties of AIMs we can establish localization theorems for the attractor and suggest a new approximate method for investigation of the long-time dynamics. A similar method for parabolic equations is known as a nonlinear Galerkin method. As examples we consider both dissipative perturbations of well-known integrable systems and some models of elastic solids subjected to nonconservative loads.
机译：我们针对一类非线性偏微分方程的长期行为和极限状态提出了一些最新结果，这些非线性方程是哈密顿系统的耗散扰动。与不受干扰的情况相反，对于所考虑的方程，存在有限维的整体吸引子。在一个相当统一的框架中，我们构造了无限个近似惯性流形（AIM）族，它们是系统相空间中的有限维光滑表面，其小区域吸引所有解并包含全局吸引子。利用AIM的性质，我们可以为吸引子建立定位定理，并提出一种新的近似方法来研究长期动力学。抛物线方程的类似方法称为非线性Galerkin方法。作为示例，我们同时考虑了众所周知的可积系统的耗散扰动和承受非保守载荷的某些弹性固体模型。

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