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On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold

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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.

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Authors and Affiliations

  1. Department of Mathematics, The University of York, Heslington, York, YO10 5DD, UK

    Z. Brzeźniak

  2. SAMM, EA 4543 and PMA, UMR 7599, Université Paris 1 Panthéon Sorbonne, 90 Rue de Tolbiac, Paris, 75634, Cedex 13, France

    A. Millet

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  1. Z. Brzeźniak
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  2. A. Millet
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Brzeźniak, Z., Millet, A. On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold. Potential Anal 41, 269–315 (2014). https://doi.org/10.1007/s11118-013-9369-2

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  • DOI: https://doi.org/10.1007/s11118-013-9369-2

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