非线性发展方程的自相似解  

The Self-similar Solutions for Some Nonlinear Evolution Equations

作　　者：苗长兴[1] 

机构地区：[1]北京应用物理与计算数学研究所,北京100088

出　　处：《数学进展》2004年第6期641-668,共28页Advances in Mathematics（China）

基　　金：国家重点基础研究发展规划项目(No.G1999075107);国家自然科学基金(No.19971011);中国工程物理研究院科学技术基金(No.20030656)资助.

摘　　要：本文着力于给出非线性发展方程的自相似解的一些最新的研究进展.借助于调和分析的方法(特别是利用Littlewood-Paley理论、时空估计等),通过非线性发展方程的Cauchy问题的研究来获得自相似解.主要技术是将初始状态空间X推广到非自反的Banach空间(使得X包含那些具自相似结构的初始函数),相应地将适定性中解在t=0处的连续性放宽成弱连续.另一方面,用Scaling的方法来分析时空可积空间的形式、非线性增长与空间X的选取等.这对非线性发展方程Cauchy问题的研究是至关重要的,它本质上给出了研究非线性发展方程Cauchy问题的工作空间.进而,对于自相似解的结构、自相似解作用(可以是某些整体解的大尺度极限)亦给出了一些具体的分析.This survey paper is devoted to some new progress on the self-similar solutions for some nonlinear evolution equations. By making use of harmonic analysis method, especially, Littlewood-Paley decomposition and time-space estimates, one class of self-similar solutions are obtained by studying the generalized well-posedness of the Cauchy Problem to the above nonlinear evolution equations. Comparing with classical well-posedess, we use some nonreflexive Banach spaces as initial state spaces which include the initial date with self-similar structure. weakly continuity to replace continuity at t = 0 and maintain the uniquness of solution in the definition of generalized well-posedness. One the other hand, we make a detailed analysis to the form of time-space Banach spaces, the growth of nonlinear term, initial state spaces and their relations by scaling technique. These facts are important to the study Cauchy problems of the nonlinear evolution equations and give the sutiable working-spaces so as to contruct fixed point of nonlinear mapping in the spaces. Meanwhile, we also give some analysis to the structure of self-similar solution and the applications of self-similar solution (the self-similar solution can be the large-scale behavior of some global solution to nonlinear evolution equations).

关 键 词：发展方程 CAUCHY问题 自相似解 BESOV空间 Littlewood-Paley分解 容许对 三元容许簇 时空估计 

分 类 号：O175.2[理学—数学]