Abstract
We show global well-posedness of certain type of strong-in-time and weak-in-space solutions for the Cauchy problem of the 1-dimensional nonlinear Schrödinger equation, in various cases of open sets, bounded and unbounded. These solutions do not vanish at the boundary or at infinity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
That is,
in \(\sigma \!\left( L^\infty \!\left( J;\mathcal {F}^*\right) ,L^1\!\left( J;\mathcal {F}\right) \right) \). Note that \(L^\infty \!\left( J;\mathcal {F}^*\right) \!\cong \!\left( L^1\!\left( J;\mathcal {F}\right) \right) ^*\) (see, e.g., [5] Theorem 1, Sect. \(\text {IV}.1\)). - 2.
This specific subset is an orthogonal basis of both \({H_0^1\!\left( U;{\mathbb {C}}\right) }\) and \({L^2\!\left( U;{\mathbb {C}}\right) }\).
- 3.
We can modify the reflection technique used for the proof of this result, in order to cover the case of the extension of \(H^2\)-functions. In particular, we can apply the reflection technique used for Theorem 5.19 in [1].
- 4.
For the \(H^2\)-regularity, we define \(\mathbf {v}_m^k\!:=\!\eta _k\mathcal {E}_U\mathbf {u}_m^k\), for all \({m\!\in \!\mathrm{I\!N}}\).
in References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, Oxford (2003)
Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, vol. 183. Springer, New York (2013)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Cazenave, T.: Semilinear Schrödinger Equations. Lecture Notes, vol. 10. American Mathematical Society, Providence, Rhode Island (2003)
Diestel, J., Uhl, J. jr.: Vector Measures, Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence, Rhode Island (1977)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence, Rhode Island (2010)
Gallo, C.: Schrödinger group on Zhidkov spaces. Adv. Differ. Equ. 9(5–6), 509–538 (2004)
Gallo, C.: The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity. Commun. Part. Differ. Equ. 33(5), 729–771 (2008)
Gérard, P.: The Cauchy problem for the Gross-Pitaevskii equation. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 23(5), 765–779 (2006)
Gialelis, N., Stratis, I.G.: Non-vanishing at spatial extremity solutions of the defocusing non-linear Schrödinger equation. Math. Methods Appl. Sci. 1–18. https://doi.org/10.1002/mma.5074
Temam, R.: Navier-Stokes Equations. Studies in Mathematics and its Applications, vol. 2, revised edn. North Holland, Amsterdam-New York (1979)
Temam, R.: Infinite-dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (2012)
Zhidkov P.E.: The Cauchy problem for a nonlinear Schrödinger equation, JINR Communications Dubna, R5-87-373, (1987) (18 pages) (in Russian)
Zhidkov, P.E.: Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lecture Notes in Mathematics, vol. 1756. Springer, Berlin-Heidelberg (2001)
Acknowledgements
– N. G. acknowledges that this research has been co-financed—via a programme of State Scholarships Foundation (IKY)—by the European Union (European Social Fund—ESF) and Greek national funds through the action entitled “Strengthening Human Resources Research Potential via Doctorate Research” (contract number: 2016–\(E\!\Sigma \!\Pi \! A\)–050-0502-5534) in the framework of the Operational Program “Human Resources Development Program, Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) 2014–2020.
– I. G. S. acknowledges that this work was made possible by NPRP grant #[8-764-160] from Qatar National Research Fund (a member of Qatar Foundation).
– The findings achieved herein are solely the responsibility of the authors.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Gialelis, N., Stratis, I.G. (2019). On the 1-dim Defocusing NLS Equation with Non-vanishing Initial Data at Infinity. In: Karapetyants, A., Kravchenko, V., Liflyand, E. (eds) Modern Methods in Operator Theory and Harmonic Analysis. OTHA 2018. Springer Proceedings in Mathematics & Statistics, vol 291. Springer, Cham. https://doi.org/10.1007/978-3-030-26748-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-26748-3_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26747-6
Online ISBN: 978-3-030-26748-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)