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Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 131–160 | Cite as

Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrödinger System in ℝ3

  • Zaihui GanEmail author
  • Xin Jiang
  • Jing Li
Article

Abstract

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schrödinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.

Keywords

Nonlocal nonlinear Schrödinger system Sharp threshold Blow-up Global existence 

2000 MR Subject Classification

35A15 35E55 35Q55 

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Notes

Acknowledgements

The authors wish to thank annoymous referees for their constructive suggestions and helpful comments.

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Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinChina
  2. 2.School of Preparatory EducationXinjiang Normal UniversityUrumqiChina

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