In this talk, I will discuss the convergence problem for the intermediate long wave equation (ILW) from deterministic and statistical viewpoints. ILW models the internal wave propagation of the interface in two-layer fluid of finite depth, providing a natural connection between the Korteweg-de Vries equation (KdV) in the shallow-water limit and the Benjamin-Ono equation (BO) in the deep-water limit.  In the first part of this talk, I discuss the convergence problem for ILW in the low regularity setting from a deterministic viewpoint. In particular, by establishing a uniform (in depth) a priori bound, I show that a solution to ILW converges to that to KdV (and to BO) in the shallow-water limit (and in the deep-water limit, respectively).  In the second part of this talk, I discuss an analogous convergence result from a statistical viewpoint. More precisely, I study convergence of invariant Gibbs dynamics for ILW in the shallow-water and deep-water limits. After a brief review on the construction of the Gibbs measure for ILW, I show that the Gibbs measures for ILW converge in total variation to that for BO in the deep-water limit, while in the shallow-water limit, we can only show weak convergence of corresponding Gibbs measures for ILW to that for KdV. In terms of dynamics, we use a compactness argument to construct invariant Gibbs dynamics for ILW (without uniqueness) and show that they converge to invariant Gibbs dynamics for KdV and BO in the shallow-water and deep-water limits, respectively.  The second part of the talk is based on joint work with Tadahiro Oh (The University of Edinburgh) and Guangqu Zheng (University of Liverpool).

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