Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

Chapouto, Andreia; Kishimoto, Nobu

doi:10.1090/tran/8699

Cited by 3 publications

(4 citation statements)

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“…Lastly, let us point out that, as for the gKdV equation (1.19) (and also (1.58)), there is a good well-posedness theory with the Gibbsian initial data; see [12,73,21]. In particular, in a recent work [21], Chapouto and Kishimoto completed the program initiated by Bourgain [13] on the construction of invariant Gibbs dynamics for the (defocusing) gKdV (1.19) for any k ∈ 2N + 1.…”

Section: Construction and Convergence Of Gibbs Measures Consider A Fi...mentioning

confidence: 99%

“…denote the Wick power defined in (1.55), 21 where σ δ,N is as in (1.54). Then, the truncated Gibbs measure ρ δ,N in (1.56) can be written as…”

Section: 2mentioning

confidence: 99%

“…In particular, the rate of convergence is uniform in 0 < δ ≤ 1. 21 As in Section 3, we freely interchange the representations in terms of X δ and in terms of v distributed by µ δ , when there is no confusion.…”

Section: 2mentioning

confidence: 99%

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On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

Oh¹,

Zheng²

2022

Preprint

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We study convergence problems for the intermediate long wave equation (ILW), with the depth parameter δ > 0, in the deep-water limit (δ → ∞) and the shallow-water limit (δ → 0) from a statistical point of view. In particular, we establish convergence of invariant Gibbs dynamics for ILW in both the deep-water and shallow-water limits. For this purpose, we first construct the Gibbs measures for ILW, 0 < δ < ∞. As they are supported on distributions, a renormalization is required. With the Wick renormalization, we carry out the construction of the Gibbs measures for ILW. We then prove that the Gibbs measures for ILW converge in total variation to that for the Benjamin-Ono equation (BO) in the deep-water limit (δ → ∞). In the shallow-water regime, after applying a scaling transformation, we prove that, as δ → 0, the Gibbs measures for the scaled ILW converge weakly to that for the Korteweg-de Vries equation (KdV). We point out that this second result is of particular interest since the Gibbs measures for the scaled ILW and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are equivalent).In terms of dynamics, we use a compactness argument to construct invariant Gibbs dynamics for ILW (without uniqueness). Furthermore, we show that, by extracting a sequence δm, this invariant Gibbs dynamics for ILW converges to that for BO in the deep-water limit (δm → ∞) and to that for KdV (after the scaling) in the shallow-water limit (δm → 0), respectively.Lastly, we point out that our results also apply to the generalized ILW equation in the defocusing case, converging to the generalized BO in the deep-water limit and to the generalized KdV in the shallow-water limit. In the non-defocusing case, however, our results can not be extended to a nonlinearity with a higher power due to the non-normalizability of the corresponding Gibbs measures.

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Section: Construction and Convergence Of Gibbs Measures Consider A Fi...mentioning

confidence: 99%

“…denote the Wick power defined in (1.55), 21 where σ δ,N is as in (1.54). Then, the truncated Gibbs measure ρ δ,N in (1.56) can be written as…”

Section: 2mentioning

confidence: 99%

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On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

Oh¹,

Zheng²

2022

Preprint

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show abstract

“…This issue necessitates a probabilistic (local) well-posedness theory, that goes beyond deterministic results by exploiting cancellations due to random oscillations. The literature on the study of Gibbs measures for nonlinear Hamiltonian PDEs is by now quite long, so we point the reader to the papers [85,80,82,83,52,51,22,27,24,75,67,59,39,76,14,26,9] and the references contained therein.…”

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confidence: 99%

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Forlano¹,

Tolomeo²

2022

Preprint

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We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter α > 1, subject to a Gaussian random initial data of negative Sobolev regularity σ < s − 1 2 , for s ≤ 1 2 . We show that for all s * (α) < s ≤ 1 2 , the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on H s (T) is quasi-invariant under the flow of the equation. For α < 1 20 (17 + 3 √ 21) ≈ 1.537, the regularity of the initial data is lower than the one provided by the deterministic wellposedness theory.We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.

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Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

Bringmann,

Deng,

Nahmod

et al. 2024

Invent. math.

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Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

Cited by 3 publications

References 50 publications

On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

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