题目：Instability and exponential dichotomy of Hamiltonian PDEs

报告人：曾崇纯 教授

（Georgia Institute of Technology，南开大学）

摘要：In this talk, we start with a general linear Hamiltonian system $u_t="JLu$" in a Hilbert space $X$ -- the energy space. We assume that (a) $J: X^* /supset D(J) /to X$ is anti-self-adjoint and (b) $L : X/to X^*$ is bounded, symmetric, with closed range $R(L)$, and its induced 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 generalized eigenspaces of eigenvalues of $JL$, some of which may be embedded in the continuous spectrum. In addition, for each eigenvalue $/lambda$ 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. Finally we will 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.

时间：7月3日（周五）上午10:00-11:00

地点：首都师大北一区文科楼 707 教室

欢迎全体师生积极参加！