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 {mathbb {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.
Highlights
Where ξ(t, x) denotes a (Gaussian) space–time white noise on R × Rd and φ is a bounded operator on L2(Rd )
Our main goal is to establish global well-posedness of (1.1) in the energysubcritical case with a rough noise, namely, with a noise not belonging to the energy space H 1(Rd )
The I -method has been applied to a wide class of dispersive models in establishing global well-posedness below the energy spaces, where there is no a priori bound on relevant norms directly given by a conservation law
Summary
We consider the Cauchy problem for the stochastic nonlinear Schrödinger equation (SNLS) with an additive noise:. Assuming φ ∈ HS(L2; H 1), de Bouard and Debussche [15] proved global well-posedness of (1.1) in H 1(Rd ) by applying Ito’s lemma to the mass M(u) and the energy E(u) in (1.5) and establishing an a priori H 1-bound of solutions to (1.1). The I -method has been applied to a wide class of dispersive models in establishing global well-posedness below the energy spaces (or more generally below regularities associated with conservation laws), where there is no a priori bound on relevant norms (for iterating a local-in-time argument) directly given by a conservation law. Our strategy for proving global well-posedness of SNLS (1.1) when φ ∈ HS(L2; H s), s < 1, is to implement the I -method in the stochastic PDE setting This will provide a general framework for establishing global well-posedness of stochastic dispersive equations with additive noises below energy spaces
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