# Analysis Seminar

#### Instability, Index Theorems, and Exponential Dichotomy of Hamiltonian PDEs

Speaker: Chongchun Zeng, Georgia Tech

Location: Warren Weaver Hall 1302

Date: Thursday, April 6, 2017, noon

Synopsis:

Motivated by the stability/instability analysis of coherent states (standing waves, traveling waves, etc.) in nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, we consider a general linear Hamiltonian system $$u_t = JL u$$ in a real Hilbert space $$X$$ -- the energy space. The main assumption is that the energy functional $$\frac 12 \langle Lu, u\rangle$$ has only finitely many negative dimensions -- $$n^-(L) < \infty$$. Our first result is an $$L$$-orthogonal decomposition of $$X$$ into closed subspaces so that $$JL$$ has a nice structure. Consequently, we obtain an index theorem which relates $$n^-(L)$$ and the dimensions of subspaces of generalized eigenvectors of some eigenvalues of $$JL$$, along with some information on such subspaces. Our third 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. Next we consider the robustness of the stability/instability under small Hamiltonian perturbations. In particular, we give a necessary and sufficient condition on whether a purely imaginary eigenvalues may become hyperbolic under small perturbations. Finally we revisit some nonlinear Hamiltonian PDEs. This is a joint work with Zhiwu Lin.