TY - JOUR
T1 - Almost conservation laws for stochastic nonlinear Schrödinger equations
AU - Cheung, Kelvin
AU - Li, Guopeng
AU - Oh, Tadahiro
N1 - Funding Information:
K.C. and G.L. were supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (Grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. K.C. and G.L. also acknowledge support from the European Research Council (grant no. 637995 “ProbDynDispEq”). T.O. was supported by the European Research Council (grant no. 637995 “ProbDynDispEq” and grant no. 864138 “SingStochDispDyn”). The authors would like to thank the anonymous referee for helpful comments.
Publisher Copyright:
© 2021, The Author(s).
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/6
Y1 - 2021/6
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 - http://www.scopus.com/inward/record.url?scp=85099587697&partnerID=8YFLogxK
U2 - 10.1007/s00028-020-00659-x
DO - 10.1007/s00028-020-00659-x
M3 - Article
AN - SCOPUS:85099587697
SN - 1424-3199
VL - 21
SP - 1865
EP - 1894
JO - Journal of Evolution Equations
JF - Journal of Evolution Equations
IS - 2
ER -