Applied Math And Analysis Seminar
Monday, October 23, 2017, 3:30pm, 227 Physics
Chongchun Zeng (Georgia Institute of Technology)
Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs

Abstract:

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.

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