偏微分方程的拟周期解

偏微分方程的KAM理论近来有了一些新的发展，例如：.1.Eliasson-Kuksin(2010,Ann.Math.)证明了高维薛定谔方程在有外参数的情况下拟周期解的存在性及线性稳定性。.2.Geng-Xu-You(2011,Adv.Math.)证明了不带外参数的二维薛定谔方程在零动量条件下拟周期解的存在性。.3.Baldi-Berti-Haus-Montalto(2018,Invent.Math.)证明了二维水波方程拟周期解的存在性及线性稳定性。..但仍有一些问题待解决，例如：.1.不带外参数的二维及二维以上薛定谔方程在没有零动量条件下是否存在拟周期解？若存在，拟周期解是否线性稳定或部分双曲的？.2.三维及三维以上水波方程是否存在拟周期解？若存在，拟周期解是否线性稳定或部分双曲的？.3.刘维尔频率的偏微分方程是否存在拟周期解？

There have been some development in KAM theory for partial differential equations,for example:.1.Eliasson-Kuksin(2010,Ann.Math.)proved that higher dimensional Schrodinger equations admit the existence of quasi-periodic solutions and their linear stability in the case of external parameters..2.Geng-Xu-You(2011,Adv.Math.)proved that two dimensional Schrodinger equations without external parameters admit the existence of quasi-periodic solutions in the case of zero momentum condition..3.Baldi-Berti-Haus-Montalto(2018,Invent.Math.)proved that two dimensional water wave equations admit the existence of quasi-periodic solutions and their linear stability..There are still some unsolved problems, for example:.1.Whether are there the existence and linear stability of quasi-periodic solutions for two dimensional and higher dimensional Schrodinger equations without extertnal parameters or zero momentum condition?.2.Whether are there the existence and linear stability of quasi-periodic solutions for three dimensional and higher dimensional water wave equations?.3.whether are there the existence of quasi-periodic solutions for partial differential equations with Liouvillean frequency?