Corrigendum: Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation (2022 Nonlinearity 
               
                  35 2878)

Fukuizumi, Reika; Hoshino, Masato; Inui, Takahisa

doi:10.1088/1361-6544/ac8c7a

Cited by 1 publication

(2 citation statements)

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“…Remark 1.14. There are recent works [34,88,89] on convergence of stochastic dynamics at the Gibbs equilibrium. One key difference between our work and these works is that, in [34,88,89], a single Gibbs measure remains invariant for the entire one-parameter family of dynamics, whereas, in our work, the Gibbs measure (and even the base Gaussian measure) varies as the depth parameter δ changes, requiring us to first establish the convergence at the level of the Gibbs measures.…”

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

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%

“… 34). In fact, the definition (2.34) is independent of the choice of a probability measure λ such that µ, ν ≪ λ.…”

mentioning

confidence: 99%

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

Oh¹,

Zheng²

2022

Preprint

View full text Add to dashboard Cite

show abstract

Corrigendum: Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation (2022 Nonlinearity 35 2878)

Cited by 1 publication

References 1 publication

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

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