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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received in December 1998

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, Trudy 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 Trudy 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. Wei J., Duan J., Gao H., Lv G., “Stochastic Regularization For Transport Equations”, Stoch. Partial Differ. Equ.-Anal. Comput.  crossref  isi
    2. Huang G., Kuksin S., “On the Energy Transfer to High Frequencies in the Damped/Driven Nonlinear Schrodinger Equation”, Stoch. Partial Differ. Equ.-Anal. Comput.  crossref  isi
    3. A. R. Shirikyan, “Analyticity of solutions for randomly forced two-dimensional Navier–Stokes equations”, Russian Math. Surveys, 57:4 (2002), 785–799  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Rougemont J., “Space–time invariant measures, entropy, and dimension for stochastic Ginzburg–Landau equations”, Communications in Mathematical Physics, 225:2 (2002), 423–448  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. 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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. 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  crossref  mathscinet  zmath  isi  scopus
    7. Kuksin S., Zhang H., “Exponential Mixing For Dissipative Pdes With Bounded Non-Degenerate Noise”, Stoch. Process. Their Appl., 130:8 (2020), 4721–4745  crossref  isi
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