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 Tr. Mat. Inst. Steklova, 1999, Volume 225, Pages 232–256 (Mi tm723)

Stochastic Nonlinear Schrödinger Equation. 1. A priori Estimates

S. B. Kuksin

Department of Mathematics, Heriot Watt University

Abstract: We consider a nonlinear Schrödinger equation with a small real coefficient $\delta$ in front of the Laplacian. The equation is forced by a random forcing that is a white noise in time and is smooth in the space-variable $x$ from a unit cube; Dirichlet boundary conditions are assumed on the cube's boundary. We prove that the equation has a unique solution that vanishes at $t=0$. This solution is almost certainly smooth in $x$, and the $k$th moment of its $m$th Sobolev norm in $x$ is bounded by $C_{m,k}\delta^{-km-k/2}$. The proof is based on a lemma that can be treated as a stochastic maximum principle.

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English version:
Proceedings of the Steklov Institute of Mathematics, 1999, 225, 219–242

Bibliographic databases:
UDC: 519.21+517.9

Citation: S. B. Kuksin, “Stochastic Nonlinear Schrödinger Equation. 1. A priori Estimates”, Solitons, geometry, and topology: on the crossroads, Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov, Tr. Mat. Inst. Steklova, 225, Nauka, MAIK «Nauka/Inteperiodika», M., 1999, 232–256; Proc. Steklov Inst. Math., 225 (1999), 219–242

Citation in format AMSBIB
\Bibitem{Kuk99} \by S.~B.~Kuksin \paper Stochastic Nonlinear Schr\"odinger Equation. 1.~A~priori Estimates \inbook Solitons, geometry, and topology: on the crossroads \bookinfo Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov \serial Tr. Mat. Inst. Steklova \yr 1999 \vol 225 \pages 232--256 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm723} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1725943} \zmath{https://zbmath.org/?q=an:0984.60070} \transl \jour Proc. Steklov Inst. Math. \yr 1999 \vol 225 \pages 219--242 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. R. Shirikyan, “Analyticity of solutions for randomly forced two-dimensional Navier–Stokes equations”, Russian Math. Surveys, 57:4 (2002), 785–799
2. Rougemont J., “Space–time invariant measures, entropy, and dimension for stochastic Ginzburg–Landau equations”, Communications in Mathematical Physics, 225:2 (2002), 423–448
3. Kuksin S., Shirikyan A., “Randomly forced CGL equation: stationary measures and the inviscid limit”, Journal of Physics A–Mathematical and General, 37:12 (2004), 3805–3822
4. Wei J., Duan J., Lv G., “Schauder Estimates For Stochastic Transport-Diffusion Equations With Levy Processes”, J. Math. Anal. Appl., 474:1 (2019), 1–22
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