Stochastic nonlinear Schrödinger equations. (May 2016)

• Record Type: Journal Article
• Authors: Barbu, Viorel; Röckner, Michael; Zhang, Deng
• Is Part Of: Nonlinear analysis. Volume 136(2016)
• Page Start: 168
• Page End: 194
• Publication Date: 2016-05
• Subjects: 60H15 -- 35B65 -- 35J10; (Stochastic) Nonlinear Schrödinger equation -- Wiener process -- Sobolev space -- Strichartz estimates; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.02.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is devoted to the well-posedness of stochastic nonlinear Schrödinger equations in the energy space H 1 ( R d ), and is a continuation of our recent work (Barbu et al., 2014). We allow general complex coefficients in the noise, thus covering both the conservative and non-conservative case, the latter being important to include quantum measurement effects in the underlying physical model. We consider both focusing and defocusing nonlinearities and prove global well-posedness in H 1 ( R d ), including also the pathwise continuous dependence on initial conditions, with exponents of the nonlinearity in exactly the same (sharp) range as in the deterministic case. In particular, in the special conservative case, this work improves earlier results in de Bouard and Debussche (2003), where the critical exponents could not be reached for d ≥ 6 in the defocusing and for d ≥ 4 in the focusing case respectively. Moreover, the local existence, uniqueness and blowup alternative are also established for the energy-critical case. The approach presented here is mainly based on the rescaling approach already used in Barbu et al. (2014) to study the L 2 case and also on the Strichartz estimates established in Marzuola et al. (2008) for large perturbations of the Laplacian.