On the Cauchy problem for focusing and defocusing Gross-Pitaevskii
hierarchies

Chen, Thomas; Pavlović, Nataša

doi:10.3934/dcds.2010.27.715

Cited by 72 publications

(192 citation statements)

References 15 publications

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“…The next step is therefore to implement moves KM (8,9), KM (7,8) and KM(6, 7), which brings the 𝜇 table to the following: This is followed by the moves KM (11,10), KM (10,9), KM (9,8) and KM (8,7), which bring the 𝜇 table to the following: At this point, the tree takes the form as pictured to the left. All 5s have been moved to their proper position in the 𝜇 table.…”

Section: Reducing To Tamed Forms Via the Signed Km Board Gamementioning

confidence: 99%

“…We have kept the 𝑡 7 integral together with 𝑡 6 because 𝐷 (7) is a child of 𝐷 (6) in the D-tree. We can see that all the Duhamel structures are fully compatible with the U-V techniques.…”

Section: − 9+mentioning

confidence: 99%

“…T. Chen and Pavlović also initiated the study of the well-posedness theory of equation (1.2) with general initial datum as an independent subject away from the quantum N -body dynamics in [7, 9, 10] (see also [12, 13, 55, 53, 54, 58, 59]). On the one hand, generalising the problem could help to attack the Klainerman–Machedon Strichartz-type bound problem.…”

Section: Introductionmentioning

confidence: 99%

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Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Chen

Holmer

2022

Forum of Mathematics, Pi

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We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

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Section: Reducing To Tamed Forms Via the Signed Km Board Gamementioning

confidence: 99%

Section: − 9+mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Chen

Holmer

2022

Forum of Mathematics, Pi

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“…They also showed that the 2D quintic case, which is usually considered the same as the 3D cubic case, satisfied the KM space-time bound while it was still open for the 3D cubic case at that time. To attack the problem, they also considered the well-posedness theory with more general data in [6,8,10]. (See also [11,51,52,53,58,59]).…”

Section: Introductionmentioning

confidence: 99%

The unconditional uniqueness for the energy-supercritical NLS

Chen,

Shen,

Zhang

2021

Preprint

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We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the R d and T d energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for H 1 -critical and H 1 -supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54] is the only possible C([0, T ); Ḣsc ) solution if exist in these domains.

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“…[37,39,33,34,35,1,2,4,5,6,7,8,20,22,23,24,21,10,18,26,28,36,31,30,25,29,32,38,40], and references therein. A fundamental problem is to prove that Bose-Einstein condensation occurs for such systems.…”

mentioning

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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On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies

Cited by 72 publications

References 15 publications

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

The unconditional uniqueness for the energy-supercritical NLS

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

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