Unconditional uniqueness for periodic nonlinear dispersive equations
主 题: Unconditional uniqueness for periodic nonlinear dispersive equations
报告人: Nobu Kishimoto (Kyoto University)
时 间: 2014-09-24 10:00-11:00
地 点: 理科一号楼1309（主持人：郭紫华）
We consider the unconditional uniqueness (UU) of solutions to the Cauchy problem for certain nonlinear dispersive equations on the torus. Our proof of UU is based on successive time-averaging arguments (integration by parts with respect to time variable, or "normal form reduction"). This approach was taken by Babin, Ilyin, and Titi (2011) for the periodic KdV equation, and has been applied to other equations such as the modified KdV equation and higher-order KdV-type equations. Recently, Guo, Kwon, and Oh (2013) obtained the optimal UU result for one-dimensional cubic NLS equation. We note that they needed to apply integration by parts infinitely many times, while for the KdV and the modified KdV cases the optimal results were obtained by finitely many applications of integration by parts. In this talk we give an abstract framework for proving UU by such an infinite iteration of integration by parts. As an application, we obtain some UU results for general NLS equations in higher dimensions and of higher (odd) degree nonlinearities and one-dimensional cubic derivative NLS. If time permits we also apply this method to the modified Benjamin-Ono equation.