Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy

Chen, Thomas; Pavlović, Nataša; Tzirakis, Nikolaos

doi:10.1090/conm/581/11491

Cited by 16 publications

(16 citation statements)

References 51 publications

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“…13 We remark that, though L (k) 3 kS(k) , it is not true that L (k) C k S (k) for any C independent of ω because of the ground state case. 14 We remind the reader that this V N ω is different from V N,ω defined in (17).…”

Section: 1mentioning

confidence: 95%

“…In another part, they show that hierarchy (14) has a unique solution which is therefore a completely factorized state. However, the uniqueness theory for hierarchy (14) is surprisingly delicate due to the fact that it is a system of infinitely many coupled equations over an unbounded number of variables. In [46], by assuming a space-time bound on the limit of γ (k) N , Klainerman and Machedon gave another uniqueness theorem regarding (14) through a collapsing estimate originating from the multilinear Strichartz estimates and a board game argument inspired by the Feynman graph argument in [31].…”

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confidence: 97%

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Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

Chen

Holmer

2017

Analysis & PDE

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We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a 3β−1 V (a β ·) where V 0 and a matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N -body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to ∞. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

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Section: 1mentioning

confidence: 95%

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confidence: 97%

Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

Chen

Holmer

2017

Analysis & PDE

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“…[18], who proved the effectiveness of the 3D to 2D reduction problem. In [12,13], Chen, Pavlović and Tzirakis worked out the virial and Morawetz identities for hierarchy (1.11) and showed the blow up for hierarchy (1.11) in 2D and 3D in the case of negative energy initial data and negative b 0 . In [29], Gressman, Sohinger, and Staffilani have obtained a uniqueness theorem of solution to hierarchy (1.11) in 3D subject to periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation

Chen

Holmer

2016

Arch Rational Mech Anal

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We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N β−1 V (N β ·) where V 0. We develop new techniques in treating the N −body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N → ∞ limit of the N -body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.

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“…The first rate of convergence result was obtained by Rodnianski and Schlein [145] and subsequent rate of convergence results have been obtained in [6,17,38,49,50,75,85,[93][94][95]110,115,121,124,137,138]. The Cauchy problem associated to a hierarchy as in (1) has been studied in its own right in [41][42][43][44]46,47,59]. A new unconditional uniqueness result for the cubic Gross-Pitaevskii hierarchy on R 3 , based on the Quantum de Finetti theorem has recently been obtained by Chen, Hainzl, Pavlović, and Seirenger in [39].…”

Section: Previously Known Resultsmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy

Cited by 16 publications

References 51 publications

Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation

Randomization and the Gross–Pitaevskii Hierarchy

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