In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schro\dinger equation with various (deterministic and random) initial data. 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 [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst. , 41 (2021), pp. 3973--3984] 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 [C. Fan and Z. Zhao, Proc. Amer. Math. Soc. , 151 (2023), pp. 2527--2542] for the necessity of the L 1 -data assumption).

Publication:

SIAM Journal on Mathematical Analysis Volume 56, Issue 3 May 3, 2024

 https://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

Email: gigliola@math.mit.edu

Zehua Zhao

Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China, andKey Laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, Beijing, China

Email: zzh@bit.edu.cn