Invariance of the Gibbs measure for the periodic quartic gKdV

Richards, Geordie

doi:10.1016/j.anihpc.2015.01.003

Cited by 35 publications

(47 citation statements)

References 41 publications

(53 reference statements)

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“…Among the reasons to reconsider Albeverio-Cruzeiro theory today, there is the clear success of randomization of initial conditions in solving dispersive equations, see for instance [10], [11], [12], [33], [34] [31] (the last two, for instance, describe another PDE that leaves a Gaussian measure invariant) and in particular the review of N. Tzvetkov [42] on nonlinear wave equation where Theorems 2.6, 2.7 are devoted to prove that solutions with poor regularity (constructed for a.e. initial condition with respect to a Gaussian measure) are the limit of more regular solutions belonging to the classical theory.…”

Section: Introductionmentioning

confidence: 99%

Weak vorticity formulation of 2D Euler equations with white noise initial condition

Flandoli

2018

Communications in Partial Differential Equations

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The 2D Euler equations with random initial condition distributed as a certain Gaussian measure are considered. The theory developed by S. Albeverio and A.-B. Cruzeiro in [1] is revisited, following the approach of weak vorticity formulation. A solution is constructed as a limit of random point vortices. This allows to prove that it is also limit of L ∞ -vorticity solutions. The result is generalized to initial measures that have a continuous bounded density with respect to the original Gaussian measure.

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Section: Introductionmentioning

confidence: 99%

Weak vorticity formulation of 2D Euler equations with white noise initial condition

Flandoli

2018

Communications in Partial Differential Equations

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“…Theorem 1.1. Assume one of the following conditions: 4 Strictly speaking, the invariance of the Gibbs measure in [35] was shown only for the gauged quartic gKdV. See Remark 1.3 below.…”

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confidence: 99%

On invariant Gibbs measures for the generalized KdV equations

Richards

Thomann

2016

Dynamics of Partial Differential Equations

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Abstract. We consider the generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.

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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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Invariance of the Gibbs measure for the periodic quartic gKdV

Cited by 35 publications

References 41 publications

Weak vorticity formulation of 2D Euler equations with white noise initial condition

Weak vorticity formulation of 2D Euler equations with white noise initial condition

On invariant Gibbs measures for the generalized KdV equations

Randomization and the Gross–Pitaevskii Hierarchy

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