学术讲座【Instability index, exponential trichotomy and invariant manifolds for Hamiltonian PDEs】

时间：2015年7月3日（星期五）下午15：00

地点：旗山校区理工北楼601报告厅

主讲：乔治亚理工学院  林治武副教授

主办：数学与计算机科学学院、数学研究中心     

专家简介：林治武，2003年获得布朗大学数学博士，现为乔治亚理工学院数学系副教授。研究方向为流体力学、非线性波、稳定性理论。在《SIAM J. Math. Anal.》、《Comm. Math. Phys.》、《Comm. Pure. Appl. Math.》等期刊上发表多篇文章。

报告摘要：Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that L induces a bounded and symmetric bi-linear form <L.,.> on X, and the energy functional <Lu,u>has only finitely many negative dimensions n(L). There is very little restriction on the anti-selfadjoint operator J, which can be unbounded and with an infinite dimensional kernel space. Our  first result is an index theorem on the linear instability of the evolution group e^tJL. More specifically, we get 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. Our second result is the linear exponential trichotomy of the evolution group e^tJL. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local variant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases, we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM, KDV and good Boussinesq equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This is a joint work with Chongchun Zeng.