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Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs
发布时间: 2017-07-10     08:36   【返回上一页】 发布人:Zeng Chongchun


      北京师范大学数学科学学院

偏微分方程系列报告

 

报告题目:Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs

报告人: Prof. Zeng Chongchun,  Georgia Institute of Technology.

时间地点:2017年712日下午10:50-11:50, 后主楼1220

邀请人:许孝精 教授  

报告摘要:We consider a general linear Hamiltonian system $u_t = JL u$ in a  Hilbert space $X$ -- the energy space. The main assumption is that the energy functional $/frac 12 /langle Lu, u/rangle$ has only finitely many  negative dimensions -- $n^-(L) < /infty$. Our first result is an  index theorem related to the linear instability of $e^{tJL}$, which  gives some relationship between $n^-(L)$ and the dimensions of spaces of generalized eigenvectors of eigenvalues of $JL$. Under some  additional non-degeneracy assumption, for each eigenvalue $/lambda /in i  R$ of $JL$ we also construct special "good" choice of generalized  eigenvectors which both realize the corresponding Jordan canonical form corresponding to $/lambda$ and work well with $L$.  Our second result is the linear exponential trichotomy of the group  $e^{tJL}$. This includes the nonexistence of exponential growth in the  finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Thirdly we  consider the structural stability of this type of systems under  perturbations if time permits. Finally we discuss applications to  examples of nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, including the construction of some local invariant  manifolds near some coherent states (standing wave, steady state,  traveling waves etc.).   This is a joint work with Zhiwu Lin.