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Long time dynamics for defocusing cubic nonlinear Schrödinger equations on three dimensional product space. (English) Zbl 1473.35520

Summary: In this article, we study long time dynamics for defocusing cubic nonlinear Schrödinger equations (NLS) on three dimensional product space. First, we apply the decoupling method in [J. Bourgain and C. Demeter, Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)] to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on a three dimensional product space with rough initial data (\(H^s\), \(s>\frac{5}{6} \)) based on the I-method and the bilinear estimate. At last, we discuss the growth of the higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35R01 PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
47A40 Scattering theory of linear operators

Citations:

Zbl 1322.42014
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Full Text: DOI arXiv

References:

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