In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.

Publication:

SIAM Journal on Mathematical Analysis, 3 May 2024

http://dx.doi.org/10.1137/23M1557544

Author:

Chenjie Fan

Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China.

Email: cjfanpku@gmail.com

Gigliola Staffilani

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Zehua Zhao

Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China.

MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing, China.