Global infinite energy solutions for the cubic wave equation

Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay

doi:10.24033/bsmf.2688

Cited by 26 publications

(55 citation statements)

References 18 publications

(41 reference statements)

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“…This is the first result on almost sure global existence of unique solutions to energycritical hyperbolic/dispersive PDEs in the periodic setting. In particular, when d = 4, Theorem 1.1 provides an affirmative answer to a question posed in [7]. When d = 4, Burq-Thomann-Tzvetkov [7] previously proved almost sure global existence (without uniqueness) of weak solutions to (1.1) on T 4 for 0 < s < 1.…”

Section: Main Resultmentioning

confidence: 87%

A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

Oh¹,

Pocovnicu²

2017

Tohoku Math. J. (2)

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Section: Main Resultmentioning

confidence: 87%

A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

Oh¹,

Pocovnicu²

2017

Tohoku Math. J. (2)

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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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“…18 See Theorems 3.1 and 3.3 for such nonlinear smoothing in the probabilistic setting. 19 Namely, the local existence time depends only on the norm of initial data.…”

Section: 2mentioning

confidence: 99%