Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Bourgain, Jean

doi:10.1007/bf02099556

Cited by 357 publications

(772 citation statements)

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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: 55%

“…In particular, we randomize the Fourier coefficients by multiplying them by a sequence of independent identically distributed standard Bernoulli random variables (meaning that their expected value is equal to 0 and their standard deviation is equal to 1). In the nonlinear dispersive equation literature, this idea was first applied in the work of Bourgain [22][23][24][25] on almost-sure well-posedness theory for the nonlinear Schrödinger equation in low regularities. These works build on a wide range of techniques on randomization in nonlinear dispersive equations, which were first developed in the work of Lebowitz, Rose and Speer [114], and Zhidkov [165].…”

Section: Setup Of the Problemmentioning

confidence: 99%

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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: 55%

Section: Setup Of the Problemmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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“…We decided not to pursue this issue here since our main concern in the present paper is to establish random data Cauchy theory for supercritical problems. We refer to [2,3,7,10,11,12] for results concerning the existence of invariant Gibbs measures in the closely related context of the Nonlinear Schrödinger equation.…”

Section: Particular Since S(t) Is 2 Periodic and Thanks To The Stricmentioning

confidence: 99%

“…The goal of this article is to show that in a very particular case we can combine this local theory with some invariant measure arguments (see the work by Bourgain [2,3] and the authors [10,11,5]) to obtain global solutions. Namely, we shall consider the nonlinear wave equation with Dirichlet boundary condition posed on Θ, the unit ball of R 3 , (1.1) (∂ 2 t − ∆)w + |w| α w = 0, (w, ∂ t w)| t=0 = (f 1 , f 2 ), u | Rt×∂Θ = 0, α > 0 with radial real valued initial data (f 1 , f 2 ).…”

Section: Introductionmentioning

confidence: 99%

Random data Cauchy theory for supercritical wave equations II: a global existence result

2008

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Abstract. -We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in ∩ s<1/2 H s (Θ). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.

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“…Of our particular interest is the statistical-mechanics aspects, which has been well studied theoretically [23][24][25]. Our goal is to predict the equilibrium statistical behavior of the low-order wave numbers.…”

Section: Introductionmentioning

confidence: 99%

Parametric reduced models for the nonlinear Schrödinger equation

Harlim

2015

Phys. Rev. E

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Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular,we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parameterization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

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Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Cited by 357 publications

References 5 publications

Randomization and the Gross–Pitaevskii Hierarchy

Randomization and the Gross–Pitaevskii Hierarchy

Random data Cauchy theory for supercritical wave equations II: a global existence result

Parametric reduced models for the nonlinear Schrödinger equation

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