Almost conservation laws for stochastic nonlinear Schrödinger equations

Kelvin Cheung, Guopeng Li, Tadahiro Oh

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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 languageEnglish
Pages (from-to)1865-1894
Number of pages30
JournalJournal of Evolution Equations
Volume21
Issue number2
Early online date20 Jan 2021
DOIs
Publication statusPublished - Jun 2021

Keywords

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

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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