论文标题
随机复杂的金茨堡 - 兰道方程的平均原理
Averaging principle for stochastic complex Ginzburg-Landau equations
论文作者
论文摘要
平均原理是研究具有高度振荡组件的动态系统的有效方法。在本文中，我们研究了随机复合物Ginzburg-Landau方程的三种平均原理。首先，我们证明原始方程的解会在有限间隔内收敛到平均方程的解决方案，而当初始数据相同时，时间尺度$ \ varepsilon $将变为零。其次，我们证明存在一个独特的复发解（特别是周期性，几乎是周期性的，几乎是自形等），当时间尺度较小时，在平均方程的固定解的附近的原始方程式存在。最后，我们在弱意义上建立了全球平均原理，即我们表明，原始系统的吸引子倾向于概率空间中平均方程的吸引子，因为$ \ varepsilon $趋于零。
Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale $\varepsilon$ goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as $\varepsilon$ goes to zero.
