Nonlinear Schrödinger equation and the twisted Laplacian-global well posedness. (English) Zbl 1319.35242
Summary: We prove the global wellposedness for the nonlinear Schrödinger equation for the magnetic Laplacian corresponding to the constant magnetic field. The magnetic Laplacian in this case can be identified with the twisted Laplacian on \(\mathbb C^n\) and our approach is via spectral theory. The local existence is established in certain first order Sobolev space, naturally associated to the twisted Laplacian, the energy space. The global well posedness is then deduced via a blowup analysis, with the help of the mass and the energy conservation established here. The general class of nonlinearities considered includes in particular, the power type nonlinearities, focussing as well as defocussing cases. The results in this paper extends the local wellposedness established in [the authors, J. Funct. Anal. 265, No. 1, 1–27 (2013; Zbl 1279.35084)].
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 42B37 | Harmonic analysis and PDEs |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B44 | Blow-up in context of PDEs |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
twisted Laplacian (special Hermite operator); nonlinear Schrödinger equation; Strichartz estimates; well posedness; conservation lawsCitations:
Zbl 1279.35084References:
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