Instability index, exponential trichotomy and invariant manifolds for Hamiltonian PDEs
主 题: Instability index, exponential trichotomy and invariant manifolds for Hamiltonian PDEs
报告人: Professor Zhiwu Lin (Georgia Institute of Technology)
时 间: 2015-06-16 16:00-17:00
地 点: 理科一号楼1493
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
on X, and the energy functional
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 invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases, we can even 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.