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Global solutions of the nonlinear Schrödinger equation with multiplicative noise

Hornung, Fabian

Abstract (englisch):
In this thesis, we investigate existence and uniqueness of solutions to the stochastic nonlinear Schrödinger equation (NLS), i.e. the NLS perturbed by a multiplicative noise.

First, we present a fixed point argument based on deterministic and stochastic Strichartz estimates. In this way, we prove local existence and uniqueness of stochastically strong solutions of the stochastic NLS with nonlinear Gaussian noise for initial values in $L^2(\mathbb{R}^d)$ and $H^1(\mathbb{R}^d),$ respectively. Using a stochastic generalization of mass conservation, we show that the $L^2$-solution exists globally under an additional restriction of the noise.

In the second part, we develop a general existence theory for global martingale solutions of the stochastic NLS with a saturated Gaussian multiplicative noise. The proof is based on a modified Galerkin approximation and a limit process due to the tightness of the approximated solutions. As an application, we get existence results for the stochastic defocusing and focusing NLS and fractional NLS on various geometries like bounded domains with Dirichlet or Neumann boundary conditions as well as compact Riemannian manifolds.
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DOI: 10.5445/IR/1000083044
Veröffentlicht am 25.05.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Publikationsjahr 2018
Sprache Englisch
Identifikator urn:nbn:de:swb:90-830442
KITopen-ID: 1000083044
Verlag KIT, Karlsruhe
Umfang VII, 210 S.
Abschlussart Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 18.04.2018
Referent/Betreuer Prof. L. Weis
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagwörter Stochastic nonlinear Schrödinger equation, multiplicative noise, Stratonovich noise, jump noise, Strichartz estimates, Galerkin method, pathwise uniqueness, martingale solutions, global solutions
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