Well-Posedness for Stochastic Nonlinear Schrödinger Equations with Rotation

Abstract

We establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with rotation in a weighted Sobolev space. In order to obtain the global well-posedness, we need some a priori estimates for the energy of the solution, which requires a careful analysis of the commutators involved. An important role is also played by the $\textrm{It}\hat{o}$ formula for the $L^p$ and Sobolev norms of the solution.

Keywords

Stochastic nonlinear Schrödinger equations / Mild/variational solutions / Strichartz estimates / Weighted Sobolev space / Energy estimates / 60H15 / 35B65 / 35J10

References

[1] AftalionAVortices in Bose–Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, 67, 2006, Boston. Birkhäuser Boston Inc.

[2] AntonelliP, MarahrensD, SparberC. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Dyn. Syst., 2012, 32(3): 703-715

[3] BarbuV, RöcknerM, ZhangD. Stochastic nonlinear Schrödinger equations. Nonlinear Anal.: Theory Methods Appl., 2016, 136: 168-194

[4] BarbuV, RöcknerM, ZhangD. The stochastic logarithmic Schrödinger equation. J. Math. Pures Appl., 2017, 107: 123-149

[5] BrzezniakZ, MilletA. On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact Riemannian manifold. Potential Anal., 2014, 41(2): 269-315

[6] BrzezniakZ, HornungF, WeisL. Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space. Probab. Theory Related Fields, 2019, 174: 1273-1338

[7] CarlesR. Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci., 2011, 9(4): 937-964

[8] Cazenave, T.: Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, Providence, RI (2003)

[9] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press (2014)

[10] de Bouard, A., Fukuizumi, R.: Representation formula for stochastic Schrödinger evolution equations and applications. Nonlinearity 25(11), 2993–3022 (2012)

[11] de BouardA, DebusscheA. A stochastic nonlinear Schrödinger equation with multiplicative noise. Commun. Math. Phys., 1999, 205(1): 161-181

[12] de BouardA, DebusscheA. The stochastic nonlinear Schördinger equation in $H^1$. Stoch. Anal. Appl., 2003, 21(1): 97-126

[13] DebusscheA, TsutsumiY. 1D quintic nonlinear Schrödinger equation with white noise dispersion. J. Math. Pures Appl. (9), 2011, 96(4): 363-376

[14] Kato, T.: Schrödinger, Nonlinear, equations, Schrödinger Operators, Sønderberg, 1988, Lecture Notes in Physics, vol. 345. Springer, Berlin 1989, 218–263

[15] LinaresF, PonceGIntroduction to Nonlinear Dispersive Equations, 2009, Springer. Universitext.

[16] MadisonKW, ChevyF, WohllebenW, DalibardJl. Vortex formation in a stirred Bose–Einstein condensate. Phys. Rev. Lett., 2000, 84(5): 806-809

[17] MadisonKW, ChevyF, WohllebenW, BretinV, DalibardJl. Stationary states of a rotating Bose–Einstein condensate: routes to vortex nucleation. Phys. Rev. Lett., 2001, 86(20): 4443-4446

[18] MatthewsMR, AndersonBP, HaljanPC, HallDS, WiemanCE, CornellEA. Vortices in a Bose–Einstein condensate. Phys. Rev. Lett., 1999, 83: 2498-2501

[19] StockS, BattelierB, BretinV, HadzibabicZ, DalibardJ. Boseeinstein condensates in fast rotation. Laser Phys. Lett., 2005, 2(6): 275-284

[20] YajimaK. Schrödinger evolution equations with magnetic fields. J. d’Anal. Math., 1991, 56(1): 29-76

[21] Zhang, D.: Stochastic nonlinear Schrödinger equation (Ph.D. thesis), Universiti at Bielefeld, (2014), http://pub.unibielefeld.de/publication/2661288

Funding

National Natural Science Foundation of China-China Academy of General Technology Joint Fund for Basic Research(11971456, 11721101)

School Start-up Fund (USTC)(KY0010000036)

the Fundamental Research Funds for the Central Universities(WK3470000016)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.