Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 2

Singular Levy processes and dispersive effects of generalized Schrödinger equations

Pages: 153 – 178

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n2.a4

Authors

Yannick Sire (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Xueying Yu (Department of Mathematics, University of Washington, Seattle, Wash., U.S.A.)

Haitian Yue (Institute of Mathematical Sciences, Shanghai Tech University, Shanghai, China)

Zehua Zhao (Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China, and the MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing, China)

Abstract

We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $\mathbb{R}^d \times \mathbb{T} , d \geq 3$. This result can be regarded as the analogue of [63, 65] in our setting and the waveguide analogue investigated in [28]. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.

Keywords

decoupling, Strichartz estimate, generalized Schrödinger equation, waveguide manifold, scattering, global well-posedness, Morawetz estimate

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35R01, 37Kxx, 37L50.

Received 27 September 2022

Published 17 May 2023