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

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JournalPotential analysis
DateE-pub ahead of print - 19 Oct 2013
DatePublished (current) - Aug 2014
Issue number2
Volume41
Number of pages47
Pages (from-to)269-315
Early online date19/10/13
Original languageEnglish

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.

    Research areas

  • Stochastic Strichartz estimates, Nonlinear Schrödinger equation, Riemannian manifold, Burkholder inequality

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