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

Abstract

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, Gérard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. 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.

References in corpus (3)

• Stochastic integration in UMD Banach spaces
• Blow-up for the stochastic nonlinear Schrodinger equation with multiplicative noise
• Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

Cited by in corpus (15)

• Scattering for stochastic nonlinear Schrödinger equations
• The Stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise
• Stochastic nonlinear Schroedinger equations in the defocusing mass and energy critical cases
• A note on the Stochastic Ericksen-Leslie equations for nematic liquid crystals
• On Global Existence and Blow-up for Damped Stochastic Nonlinear Schrödinger Equation
• The three dimensional stochastic Zakharov system
• A note on log-log blow up solutions for stochastic nonlinear Schrödinger equations
• Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case
• Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equation
• Minimal mass blow-up solutions to rough nonlinear Schroedinger equations
• Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation
• Long time behavior of stochastic NLS with a small multiplicative noise
• On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations
• The law of large numbers for quantum stochastic filtering and control of many particle systems
• Global Solution and Blow-up of the Stochastic Nonlinear Schrödinger system