The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(\mathbb{T})$ for any $s \geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.
Global Kato type smoothing estimates via local ones for dispersive equations
本文证明了包括薛定谔方程在内的一类色散方程的局部Kato型光滑估计与整体Kato型光滑估计是等价的.由此我们立即得到如下两个结果。一个是已知的局部Kato平滑估计是尖锐的。全局Kato平滑估计的尖锐正则性范围是已知的，但局部Kato平滑估计的尖锐正则性范围是未知的。最近，Sun、Trélat、Zhang和Zhong只在时空$\mathbb R \times \mathbb R$中证明了它。我们的结果在更高的维度上解决了这个问题。另一个是尖锐的全球时间最大薛定谔估计。最近，薛定谔方程的逐点收敛猜想已由杜-古思-李-张和杜-张解决.为此，他们证明了相关的尖锐的局部最大薛定谔估计。通过我们的结果，这些导致尖锐的全球时间最大薛定谔估计。
Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation
Université Paris Dauphine（巴黎第九大学）；Humboldt-Universitaet zu Berlin（柏林洪堡大学）；Institut für Mathematik（数学研究所）；The University of Edinburgh（爱丁堡大学）；Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；Institut Universitaire de France（法国高等研究院）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；Institut für Angewandte Mathematik Universität Bonn（波恩大学应用数学研究所）；University of Oxford（牛津大学）；Beijing Institute of Technology（北京理工大学）
Refined global well-posedness for the periodic modulated Korteweg-de Vries equation
The University of Edinburgh（爱丁堡大学）；The Maxwell Institute for the Mathematical Sciences（数学科学马克斯韦尔研究所）；University of Oxford（牛津大学）；University of Bonn（波恩大学）；Beijing Institute of Technology（北京理工大学）
