The Gross–Pitaevskii Hierarchy on General Rectangular Tori

Herr, Sebastian; Sohinger, Vedran

doi:10.1007/s00205-015-0950-2

Cited by 30 publications

(25 citation statements)

References 75 publications

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“…6 Later, Kirkpatrick, Schlein, and Staffilani [43] obtained the KM space-time bound via a simple trace theorem in both R 2 and T 2 and derived the 2D cubic defocusing NLS from the 2D quantum many-body dynamic. Such a scheme also motivated many works [11,13,18,20,35,38,57,58,61] for the uniqueness of GP hierarchies and enables the hierarchy method on the derivation 1D or 2D NLS directly from 3D [16,20,55], which is quite different but has some similar flavor with our Theorem 1.1 here. However, how to verify the KM bound in the 3D cubic case remained fully open at that time.…”

Section: Introductionmentioning

confidence: 77%

The derivation of the compressible Euler equation from quantum many-body dynamics

Chen¹,

Shen²,

Wu³

et al. 2021

Preprint

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We study the three dimensional many-particle quantum dynamics in meanfield setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck's constant tends to zero. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds. We have hence found a physical meaning for Strichartz type bounds which were first raised by Klainerman and Machedon in this context. Contents 1. Introduction 1.1. Statement of the Main Theorem 1.2. Outline of the Proof 2. BBGKY Hierarchy v.s. H-NLS: Long-time Uniform in Estimates 2.1. A Tool Box of Space-time Estimates 2.2. A Klainerman-Machedon Bound 1st 2.3. Feeding the Strichartz Bound into the H 1 Estimate 2.4.

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Section: Introductionmentioning

confidence: 77%

The derivation of the compressible Euler equation from quantum many-body dynamics

Chen¹,

Shen²,

Wu³

et al. 2021

Preprint

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“…Around the same time, Gressman, Sohinger and Staffilani [32] studied the uniqueness of equation (1.2) in the T 3 setting and found that the sharp collapsing estimate on T 3 needs 𝜀 more derivatives than the R 3 case, in which one derivative is needed. Herr and Sohinger later generalised this fact to all dimensions [34] -that is, collapsing estimates on T 𝑛 always need 𝜀 more derivatives than the R 𝑛 case proved in [16]. 8 In 2013, T. Chen, Hainzl, Pavlović and Seiringer introduced the quantum de Finetti theorem, from [51], to the derivation of the time-dependent power-type NLS and provided, in [6], a simplified proof of the R 3 unconditional uniqueness theorem regarding equation (1.2) from [29].…”

Section: Forum Of Mathematics Pimentioning

confidence: 90%

“…Around the same time, Gressman, Sohinger and Staffilani [32] studied the uniqueness of equation (1.2) in the setting and found that the sharp collapsing estimate on needs more derivatives than the case, in which one derivative is needed. Herr and Sohinger later generalised this fact to all dimensions [34] – that is, collapsing estimates on always need more derivatives than the case proved in [16]. 8…”

Section: Introductionmentioning

confidence: 99%

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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“…Around the same period of time, Gressman, Sohinger, and Staffilani [29] studied the uniqueness of the GP hierarchy on T 3 and proved that the sharp space-time estimate on T 3 needed an additional ε derivatives than the R 3 setting in which one derivative is needed. Later, Herr and Sohinger generalized this fact to more general cases in [32].…”

Section: Introductionmentioning

confidence: 90%

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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The Gross–Pitaevskii Hierarchy on General Rectangular Tori

Cited by 30 publications

References 75 publications

The derivation of the compressible Euler equation from quantum many-body dynamics

The derivation of the compressible Euler equation from quantum many-body dynamics

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

The unconditional uniqueness for the energy-supercritical NLS

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