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Remark on the local well-posedness for NLS with the modulated dispersion

Part of: Stochastic analysis Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  02 April 2025

Tomoyuki Tanaka Open the ORCID record for Tomoyuki Tanaka [Opens in a new window]
Tomoyuki Tanaka*
Affiliation:
Graduate School of Engineering Science, Yokohama National University, Hodogaya-ward, Yokohama, Kanagawa, 240-8501, Japan (tanaka-tomoyuki-fp@ynu.ac.jp)
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Abstract

We consider the Cauchy problem of the non-linear Schrödinger equation with the modulated dispersion and power type non-linearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk–Gubinelli [7] and multilinear estimates which are based on divisor counting and show the local well-posedness. This generalizes the result by Chouk–Gubinelli [7] in terms of the dimension and the order of the non-linearity.

Keywords

local well-posednessmodulated dispersionnon-linear Schrödinger equationYoung integralregularization by noise

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35A01: Existence problems 60H15: Stochastic partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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