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: 2016-06-14 16:00-17:00
地 点: 理科一号楼 1418
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 no restriction on the anti-selfadjoint operator J. First, we get an index theorem on the linear instability of the Hamiltonian PDE. 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. Such exponential trichotomy is important to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). We will discuss applications to Gross-Pitaveskii equation for superfluids, modulational instability for dispersive models, 2D Euler equation for ideal fluids, Vlasov-Maxwell systems for collisionless plasmas etc. This is a joint work with Chongchun Zeng.