On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Bényi, Árpád; Oh, Tadahiro; Pocovnicu, Oana

doi:10.1090/btran/6

Cited by 123 publications

(334 citation statements)

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“…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%

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: Previously Known Resultsmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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“…We should, however, point out that, with the spaces introduced by Koch and Tataru, we can also prove probabilistic small data global well-posedness and scattering as a consequence of the probabilistic global-in-time Strichartz estimates (Proposition 1.4). See our paper [4] for an example of such results for the cubic…”

Section: 4mentioning

confidence: 93%

“…While the mass of v in (1.13) has a global-in-time control, there is no energy conservation for v and thus we do not know how to proceed at this point. In [4], we establish almost sure global existence for (1.12), assuming an a priori control on the H 1 -norm of the nonlinear part v of a solution. We also prove there, without any assumption, global existence with a large probability by considering a randomization, not on unit cubes but on dilated cubes this time.…”

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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“…Such questions were further explored by Burq-Tzvetkov [17,18] in the context of the cubic nonlinear wave equation on a three-dimensional compact Riemannian manifold. There has since been a vast body of research where probabilistic tools are used to study many nonlinear dispersive or hyperbolic equations in scaling super-critical regimes, see for example [21,48,23,19,14,13,59,49,50,45,9,53,24,25] and references therein.…”

Section: Introductionmentioning

confidence: 99%

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Kenig

Mendelson

2019

International Mathematics Research Notices

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We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve inḢ s, for any s > 1/2. By randomizing radial initial data inḢ sfor s > 5/6, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.1.1. Statement of the main theorem and discussion of methods. We consider the stationary solutions φ a and we make the following ansatz:We take the scaling parameter a(t) to be time dependent, and as remarked in [40], this leads to analysis similar to modulation theory, but distinct since resonances are not associated with L 2 projections. The resonance ϕ a(t) := ∂ a φ a(t)from the definition of P ac , see (3.3) below, we see that the condition in the X s norm is weaker than requiring the same bounds for the function f itself. Moreover, this definition highlights the fact that the Lorentz condition will only be required for the low-frequency component, see (7.4).Let s ∈ R and definefor some ε < ε 0 , where ε 0 , C, c > 0 are determined in Theorem 1.5 below.Remark 1.3. In the case of finite energy initial data, the orthogonality conditionensures that the second variation of the energy restricted to a certain codimension one Lipschitz submanifold is non-negative at φ a , see the discussion following [40, Theorem 1]. Although this does not apply to our (infinite energy) initial data, more concretely, we use this condition to avoid growth when solving an ODE for the pure point component of the solution, see (7.12).

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On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Cited by 123 publications

References 56 publications

Randomization and the Gross–Pitaevskii Hierarchy

Randomization and the Gross–Pitaevskii Hierarchy

Excursions in Harmonic Analysis, Volume 4

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

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