The Application Of Normal Form And I-method In The Partial Differential Equations

In this paper,we will mainly study the global properties of cubic nonlinear Schr6dinger equations on the compact Riemannian manifold without boundary by I-method,the global solutions for nonlinear Klein-Gordon equations in infinite homogeneous waveguides,R×M,with smooth and small initial data by the normal form method,where(M,g) is a Zoll manifold or a compact revolution hypersurface, and finally,combining I-method and the resonant decomposition,which is similar to the normal form method,to deal with the L~2-concentration phenomenon for Zakharov system in R~2,with radial initial data,which are below the energy norms.First of all,we will talk about the local and global results and methods of SchrSdinger equations in the whole spaces,mainly about H~1- critical,subcritical and L~2- critical,subcritical cases,also about the L~2-concentration phenomenon for finite time blow up solutions to the L~2-critical nonlinear Schr(o|¨)dinger equations, which would give some basic ideas of this paper.Secondly,we will study the cubic nonlinear Sch(o|¨)dinger equations on the compact Riemannian manifold without boundary,mainly about:1.the estimate of the growth speed of the high Sobolev norms;2.the global existence of the solutions with the initial data below H~1 by the I-method.Thirdly,we will use the normal form method to prove the global solutions for nonlinear Klein-Gordon equations in infinite homogeneous waveguides,R×M, with smooth and small initial data,where(M,g) is a Zoll manifold or a compact revolution hypersurface.In that chapter,we will state the R~2×M case briefly.Finally,for the last chapter of this paper,we will give the proof for the L~2concentration phenomenon for Zakharov system in R~2,with radial initial data, below the energy norms,by combining the resonant decomposition,which is a little similar to the normal form method,with the I-method,and we will also introduce the global result of R case briefly.All of these would release the similar properties between Sch(o|¨)dinger equations and Zakharov systems.