Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

Silva, Daniela De; Pavlović, Nataša; Staffilani, Gigliola; Tzirakis, Nikolaos

doi:10.3934/dcds.2007.19.37

Cited by 37 publications

(49 citation statements)

References 19 publications

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“…This estimate turns out to be crucial in [Hani 2012] where it is proved that (4-7) is globally well-posed for all s > 2 3 . This generalizes, without any loss in regularity, a similar result from [Bourgain 2004] (see also [De Silva et al 2007]), where global well-posedness for s > 2 3 is proved for the torus ‫ޔ‬ 2 . Global well-posedness for s 1 follows using conservation of energy and standard arguments.…”

Section: Further Results and Remarkssupporting

confidence: 82%

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2012

Anal. PDE

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We give a detailed microlocal study of X-ray transforms over geodesic-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied.Stefanov is partly supported by NSF grant DMS-0800428. Uhlmann is partly supported by NSF FRG grant 0554571 and a Walker Family Endowed Professorship. MSC2000: 53C65.

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Section: Further Results and Remarkssupporting

confidence: 82%

“…In the context of compact manifolds, some bilinear estimates on the torus were already implicit in ] (see also [Burq et al 2005a]), and other variants were proved in [De Silva et al 2007].…”

Section: Introductionmentioning

confidence: 99%

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2012

Anal. PDE

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“…To prove Lemma 4.3, we need some knowledge from number theory, we list them below, which can be found in [19,3]. Proof of Lemma 4.3.…”

Section: Derivation Of the Quintic Nls From Many-body Quantum Dynamicmentioning

confidence: 99%

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Yuan¹

2015

Communications on Pure &Amp; Applied Analysis

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In this paper, we investigate the dynamics of a boson gas with three-body interactions in T 2 . We prove that when the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized N -body initial datum, we show that this N → ∞ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.2000 Mathematics Subject Classification. Primary: 35L15, 35L45; Secondary: 35Q40.

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“…Remark 3.6. In Silva-Pavlovic-Staffilani-Tzirakis [30], there are bilinear estimates as well and they are proved by using number theory techniques. Recent decoupling method established in Bourgain-Demeter [6] allows people to derive bilinear estimate as in Fan-Staffilani-Wang-Wilson [21].…”

Section: Bilinear Strichartz Estimatementioning

confidence: 99%

“…Additionally, we cover the 3d tori case as well (when n = 0) in (1.1). For 1d and 2d tori case, please see Silva-Pavlovic-Staffilani-Tzirakis [30].…”

Section: Introductionmentioning

confidence: 99%

Long time dynamics for defocusing cubic NLS on three dimensional product space

Zhao¹,

Zheng²

2020

Preprint

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In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter [6] to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data (H s , s > 5 6 ) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.

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Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

Cited by 37 publications

References 19 publications

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Untitled

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Long time dynamics for defocusing cubic NLS on three dimensional product space

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