Invariant measure and long time behavior of
regular solutions of the Benjamin–Ono equation

Sy, Mouhamadou

doi:10.2140/apde.2018.11.1841

Cited by 19 publications

(15 citation statements)

References 17 publications

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“…To obtain these results, an additional difficulty compared to the earlier works is the fact that we know only one coercive conservation law for KG (in the case of the torus we have also the momentum which is not coercive), that implies a "lack of estimates". Notice also that the FDL approach was developped for Hamiltonian PDEs having at least two "good" conservation laws [Kuk04,KS04,KS12,Sy16]. The present paper is also intended to extend this approach to Hamiltonian PDEs having one conservation law.…”

Section: Statement Of the Main Results And Commentsmentioning

confidence: 99%

“…This makes the Gibbs measures particularly adapted to approach some spaces of low regularity and to give a probabilistic alternative to the Cauchy theory for PDEs. However, FDL measures can approach some high regularity spaces, seemingly inaccessible by the formers, to establish long time behavior properties of PDEs (see [Sy16]). The intermediate situation is common to both.…”

Section: Introductionmentioning

confidence: 99%

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Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

2018

Stoch PDE: Anal Comp

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In this paper we construct an invariant probability measure concentrated on H 2 (K) × H 1 (K) for a general cubic Klein-Gordon equation (including the case of the wave equation). Here K represents both the 3-dimensional torus or a bounded domain with smooth boundary in R 3 . That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the Fluctuation-Dissipation-Limit (FDL) approach to PDEs having only one (coercive) conservation law.

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Section: Statement Of the Main Results And Commentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

2018

Stoch PDE: Anal Comp

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“…These estimates allow to perform a globalization argument. See [25,40,51,53] for works related to this method.…”

Section: Invariant Measure As a Tool Of Globalizationmentioning

confidence: 99%

Global well-posedness and long-time behavior of the fractional NLS

2021

Stoch PDE: Anal Comp

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In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $$\ge 2$$ ≥ 2 . In the high regularity result, an Inviscid - Infinite dimensional (IID) limit is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities.

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“…Finally, the method (iii) appears to be more flexible than the others in the context of fluid mechanics or even dispersive PDEs. We refer to [16] for applications of this method to dispersive equations and the work of Sy [23,22,24]. The draw-back of the fluctuation-dissipation method is that since the invariant measures are constructed by a compactness technique, the nature of the invariant measures is not as good as the Gaussian measures of method (ii) or (i).…”

Section: Invariant Measures For (E)mentioning

confidence: 99%

Construction of High Regularity Invariant Measures for the 2D Euler Equations and Remarks on the Growth of the Solutions

Latocca¹

2020

Preprint

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We consider the Euler equations on the torus in dimensions 2 and 3 and we construct invariant measures for the dynamics of these equations concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist at least locally. The proof follows the method of Kuksin in [17] and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures such as diracs. Then we prove that µ-almost every initial data gives rise to a global solution for which the growth of the Sobolev norms are at most polynomial. We point out that up to the knowledge of the author, the only general upper-bound for the growth of the Sobolev norm to the 2d Euler equations is double exponential. In the other direction, some initial data are known to produce solutions exhibiting exponential growth, as in the work of Zlatos [26].

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Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation

Cited by 19 publications

References 17 publications

Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3D

Global well-posedness and long-time behavior of the fractional NLS

Construction of High Regularity Invariant Measures for the 2D Euler Equations and Remarks on the Growth of the Solutions

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