某些随机非线性发展方程组的动力学行为

项目名称： 某些随机非线性发展方程组的动力学行为

项目编号： No.11301097

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 郭艳凤

作者单位： 广西科技大学

项目金额： 23万元

中文摘要： 本项目主要对某些随机非线性发展方程组的动力学行为展开研究。研究的物理模型主要包括随机古典Zakharov系统，随机量子Zakharov系统，随机大气-海洋系统，随机长短波方程等。本项目将深入讨论在有界区域和无界区域上的随机非线性发展方程的耦合方程组的随机动力系统的存在性、整体随机吸引子的存在唯一性、不变测度的存在性及其和随机吸引子之间的关系等随机动力学行为。我们将利用Sobolev空间理论、半群理论、随机微分方程理论、随机过程的遍历性等理论，并结合能量估计、紧性方法、It?公式、鞅不等式、期望意义下的积分最大值估计等估计技巧来系统的研究随机非线性发展方程的耦合方程组的随机动力学行为，从而得到相关的数学理论成果。所研究的问题不仅具有重要的理论意义，而且还具有一定的前沿性和广泛的应用价值。

中文关键词： 随机非线性发展方程组；随机动力学行为；随机动力系统；随机吸引子；不变测度

英文摘要： The project is mainly devoted to study dynamics behaviors of some stochastic nonlinear evolution equations. The models that we will investigate mainly include stochastic classical Zakharov sysetem, stochastic quantum Zakharov system, stochastic atmosphere-ocean system, stochastic long-short wave equation and so on. In this project, we will deeply study stochastic dynamics behaviors of coupled groups for nonliear evolution equations on bounded and unbounded domains, which include existence of stochastic dynamical system, existence and uniqueness of global stochastic or random attractors, existene of invariant measures, relations between invariant measures and attractors, and so on. Stochastic dynamics behaviors have been mainly investigated by using space theory of Sobolev, theory of semigroups, theory of stochastic differential equations, ergodic thoery of random processes combining with techniques of some estimates such as energy estimates, method of compactness, It? formula, martingale inequality, Maximum estimates of integral in sense of expectation. Thus the results in mathematical theories will be obtained. The contents of this project are not only important in theories but also have great advancing and values of wide applications in practices.

英文关键词： Stochastic nonlinear evolution；Stochastic dynamics behaviors；Stochastic dynamical system；Random attractors；Invariant measures