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學術活動

學術活動

Instability and exponential dichotomy of Hamiltonian PDEs

主  讲  人:曾崇纯 教授(Georgia Institute of Technology,南开大学)
时      间:2015年7月3日(周五)上午10:00-11:00
地      点:尊龙凯时北一区文科楼 707 教室
主 办 单 位:尊龙凯时数学科学学院
内 容 介 绍: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.
 

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