Analysis Seminar

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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.