scholarly journals Global Wellposedness in the Energy Space for the Maxwell-Schrödinger System

2009 ◽  
Vol 288 (1) ◽  
pp. 145-198 ◽  
Author(s):  
Ioan Bejenaru ◽  
Daniel Tataru
Keyword(s):  
Energy Space ◽  
Global Wellposedness ◽  
Schrödinger System

scholarly journals Almost conservation laws for stochastic nonlinear Schrödinger equations

2021 ◽  
Author(s):  
Kelvin Cheung ◽  
Guopeng Li ◽  
Tadahiro Oh
Keyword(s):  
Conservation Laws ◽  
Stopping Time ◽  
Energy Space ◽  
Stochastic Forcing ◽  
Time Dynamics ◽  
Nonlinear Schrödinger ◽  
Time Argument ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Stochastic Setting ◽  
Ito's Lemma

AbstractIn 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 $${\mathbb {R}}^3$$ R 3 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.

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