Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2(T)

Colliander, J.; Oh, Tadahiro

doi:10.1215/00127094-1507400

Cited by 156 publications

(304 citation statements)

References 41 publications

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“…This is similar to the idea of Wickordering [24,61] and related ideas applied in [128] in the context of the nonlinear Schrödinger equation. In fact, we will work in the non-resonant class N , which is precisely defined in Section 6.4.…”

Section: Setup Of the Problemsupporting

confidence: 54%

“…As we will see below, we will be able to prove an upper bound on (71) if we impose additional non-resonance conditions. This is reminiscent of the ideas in [23,61,128]. The precise bound is given in Proposition 6.2 below.…”

Section: A Special Class Of Density Matricesmentioning

confidence: 80%

“…We refer the interested reader to the works [18,19,[28][29][30][31][32][33][34][35][36]61,[68][69][70]122,125,126,[130][131][132][133]144,[150][151][152][153][154][155][156][157][158]163], as well as to the expository works [27,166] and to the references therein. Furthermore, we note that the idea of randomization of the Fourier coefficients without the use of an invariant measure has also been applied in the context of the Navier-Stokes equations.…”

Section: Previously Known Resultsmentioning

confidence: 99%

“…However, as is shown in Section 6.1, it is not possible to prove a good spatial estimate in the class of general density matrices by applying the combinatorial method used to prove (3). In Section 6.4, an appropriate spatial estimate is shown if one imposes an additional condition of non-resonance, similar to [24,61,128]. In fact, by working in the non-resonant classes, we are able to prove an estimate in a regime which allows us to go all the way down to the regularity of L 2 , that is to α = 0.…”

Section: Ideas and Techniques Used In The Proofsmentioning

confidence: 95%

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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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Section: Setup Of the Problemsupporting

confidence: 54%

Section: A Special Class Of Density Matricesmentioning

confidence: 80%

Section: Previously Known Resultsmentioning

confidence: 99%

Section: Ideas and Techniques Used In The Proofsmentioning

confidence: 95%

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Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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“…Remark 1.9. In probabilistic well-posedness results [6,8,20,39] for NLS on T d , random initial data are assumed to be of the following specific form: 14) where {g n } n∈Z d is a sequence of independent complex-valued standard Gaussian random variables. The expression (1.14) has a close connection to the study of invariant (Gibbs) measures and, hence, it is of importance.…”

Section: 4mentioning

confidence: 99%

Excursions in Harmonic Analysis, Volume 4

Bălan¹,

Begue²,

Benedetto³

et al. 2015

Applied and Numerical Harmonic Analysis

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Abstract. We consider a randomization of a function on R d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energycritical cubic nonlinear Schrödinger equation on R 4 is almost surely locally well-posed with respect to randomized initial data below the energy space.

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Invariant Measures and the Soliton Resolution Conjecture

Chatterjee

2013

Comm Pure Appl Math

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The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multisoliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation to date. This paper proves a “statistical version” of this conjecture at mass‐subcritical nonlinearity, in the following sense: The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long‐term behavior for “generic initial data” with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to 0. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting. The above results, following in the footsteps of a program of studying the long‐term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose, and Speer and carried forward by Bourgain, McKean, Tzvetkov, Oh, and others, are proved using a combination of techniques from large deviations, PDEs, harmonic analysis, and bare‐hands probability theory. It is valid in any dimension. © 2014 Wiley Periodicals, Inc.

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Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2(T)

Cited by 156 publications

References 41 publications

Randomization and the Gross–Pitaevskii Hierarchy

Randomization and the Gross–Pitaevskii Hierarchy

Excursions in Harmonic Analysis, Volume 4

Invariant Measures and the Soliton Resolution Conjecture

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