On the stochastic Strichartz estimates and the stochastic nonlinear Schrodinger equation on a compact riemannian manifold

Zdzislaw Brzezniak; Annie Millet

doi:10.1007/s11118-013-9369-2

On the stochastic Strichartz estimates and the stochastic nonlinear Schrodinger equation on a compact riemannian manifold

Zdzislaw Brzezniak, Annie Millet

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d = 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

Original language	English
Pages (from-to)	269-315
Number of pages	47
Journal	Potential analysis
Volume	41
Issue number	2
Early online date	19 Oct 2013
DOIs	https://doi.org/10.1007/s11118-013-9369-2
Publication status	Published - Aug 2014

Keywords

Stochastic Strichartz estimates
Nonlinear Schrödinger equation
Riemannian manifold
Burkholder inequality

Access to Document

10.1007/s11118-013-9369-2

Cite this

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title = "On the stochastic Strichartz estimates and the stochastic nonlinear Schrodinger equation on a compact riemannian manifold",

abstract = "We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr{\"o}dinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d = 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.",

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author = "Zdzislaw Brzezniak and Annie Millet",

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T1 - On the stochastic Strichartz estimates and the stochastic nonlinear Schrodinger equation on a compact riemannian manifold

AU - Brzezniak, Zdzislaw

AU - Millet, Annie

PY - 2014/8

Y1 - 2014/8

N2 - We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d = 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

AB - We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d = 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

KW - Stochastic Strichartz estimates

KW - Nonlinear Schrödinger equation

KW - Riemannian manifold

KW - Burkholder inequality

U2 - 10.1007/s11118-013-9369-2

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M3 - Article

SN - 0926-2601

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EP - 315

JO - Potential analysis

JF - Potential analysis

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