摘要
本讲义包含两部分.在第一部分中,我们的目的是证明由属于Sobolev空间的向量场生成的可测映射流的存在性,为此我们讨论常微与运输方程和连续方程的联系.第二部分处理随机微分方程:在系数不满足Lipschitz条件时,我们显式地构造出了强解;在整体Lipschitz条件下,完整地证明了随机同胚流的存在性,并讨论了它与Fokker-Planck方程的关系;最后我们强调了椭圆性在把弱解变成强解中所起的作用.
This note consisting of two parts gives a survey of new developments on the topic concerning ordinary differential equations, stochastic differential equations and partial differential equations. In the first one, our aim is to prove the existence of a flow of measurable maps associated to a vector field belonging to a Sobolev space; to this end, we discuss the link with transport equations and continuity equations. The second part deals with stochastic differential equations: we construct explicitly strong solutions beyond Lipschitz conditions, the existence of a stochastic flow of homeomorphisms under global Lipschitz conditions is fully proved and the relation with Fokker-Planck equations is discussed, and finally we emphasize the role of ellipticity to make weak solutions to strong one.
出处
《数学进展》
CSCD
北大核心
2009年第5期513-552,共40页
Advances in Mathematics(China)