Seminar Event Detail

Differential Equations

Date:  Thursday, September 24, 2015
Location:  4088 East Hall (4:00 PM to 5:00 PM)

Title:  Instability index, exponential trichotomy and invariant manifolds for Hamiltonian PDEs

Abstract:   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 <L.,.> on X, and the energy functional <Lu,u> has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J, which can be unbounded and even 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 prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and Boussinesq equations, 2D Euler equation for ideal fluids, Vlasov-Maxwell systems for collisionless plasmas etc. This is a joint work with Chongchun Zeng.


Speaker:  Zhiwu Lin
Institution:  Georgia Tech

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