Almost conservation laws for stochastic nonlinear Schrödinger equations

Kelvin Cheung; Guopeng Li; Tadahiro Oh

doi:10.1007/s00028-020-00659-x

Almost conservation laws for stochastic nonlinear Schrödinger equations

Kelvin Cheung
, Guopeng Li
, Tadahiro Oh^*

^*Corresponding author for this work

Heriot-Watt University
The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences
University of Edinburgh

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on R³ with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

Original language	English
Pages (from-to)	1865-1894
Number of pages	30
Journal	Journal of Evolution Equations
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s00028-020-00659-x
Publication status	Published - Jun 2021
Externally published	Yes

Keywords

Almost conservation law
Global well-posedness
I-method
Stochastic nonlinear Schrödinger equation

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10.1007/s00028-020-00659-x

Cite this

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title = "Almost conservation laws for stochastic nonlinear Schr{\"o}dinger equations",

abstract = "In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schr{\"o}dinger equation (SNLS) on R3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito{\textquoteright}s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.",

keywords = "Almost conservation law, Global well-posedness, I-method, Stochastic nonlinear Schr{\"o}dinger equation",

author = "Kelvin Cheung and Guopeng Li and Tadahiro Oh",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s).",

year = "2021",

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language = "English",

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T1 - Almost conservation laws for stochastic nonlinear Schrödinger equations

AU - Cheung, Kelvin

AU - Li, Guopeng

AU - Oh, Tadahiro

PY - 2021/6

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N2 - In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on R3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

AB - In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on R3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

KW - Almost conservation law

KW - Global well-posedness

KW - I-method

KW - Stochastic nonlinear Schrödinger equation

UR - https://www.scopus.com/pages/publications/85099587697

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DO - 10.1007/s00028-020-00659-x

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VL - 21

SP - 1865

EP - 1894

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 2

ER -

Almost conservation laws for stochastic nonlinear Schrödinger equations

Abstract

Keywords

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