Title: Some results on the cubic Szego equation with a linear pertubation
Speaker: Haiyan XU, Laboratoire de Mathématiques d’Orsay, Université Paris- Sud (XI), 91405, Orsay Cedex, France
E-mail address: haiyan.xu@math.u-psud.fr
Date: 2015.3.19 4:00-5:00PM
Place: 数学学院二楼学术报告厅
 
 
Abstract
We consider the following nonlinear Hamiltonian equation
$$ i u_t = Π(|u|^2 u) + α (u|1), α ∈ R,$$
where Π denotes the Szego projector on the Hardy space of the circle $S^1$. The equation with α = 0 was first introduced by Gérard and Grellier as a toy model for totally non dispersive evolution equations. For the case α≠0, the system is proved to be globally well-posed and completely integrable under the Liouville sense. Furthermore, we establish the following properties for this equation. For α < 0, any compact subset of initial data leads to a relatively compact subset of trajectories. For α > 0, there exist trajectories on which high Sobolev norms grow with time. Especially there exist solutions with their Sobolev norms exponentially growing in time. Bourgain and Staffilani has already proved the upper bound for the dispersive equations is polynomial in time, our growth is much faster. It is not too strange since our model is non dispersive, and it is the first example with such a high growth.