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08-16微分方程系列报告【Chenmin Sun】
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TitleGIBBS MEASURE FOR THE FRACTIONAL NONLINEAR SCHRöDINGER EQUATIONS
SpeakerChenmin Sun    (Universite Cergy-Pointoise)
Time2019年8月16日            下午    16:00-17:00
Room东区管理科研楼   数学科学学院1418室

AbstractWe consider the fractional nonlinear Schr¨odinger equation with cubic nonlinearity: 
                                                 $$i\partial_tu-(-\partial_x^2)^{\alpha/2}u=|u|^2u.\eqno(0.1)$$
(0.1) is a Hamiltonian system with conserved energy
                              $$H(u)=\int_{\mathbb{T}}\left({1\over 2}|D^{\alpha/2}u|^2+{1\over 4} |u|^4\right)dx.$$
The case $\alpha=2$ corresponds to the classical nonlinear Schr\¨odinger equation. I will first explain the construction of its Gibbs measure, which is formally of the form $d\mu=e^{-H(u)}du$, for the strong dispersive case $\alpha>1$. For the weak dispersive case $\alpha\leq 1$, a renormalization procedure is needed, in order to make sense of the formal expression. Next I will discuss three methods for construting global dynamics on the support of the Gibbs measure, according to the value of $\alpha$. This talk is based on a joint work with N. Tzvetkov.


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