Deep-water and shallow-water limits of the intermediate long wave equation: from deterministic and statistical viewpoints

Abstract

In this thesis, we study 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 a 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 thesis, we 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, we show that a solution to ILW converges to that to KdV (and to BO) in the shallow-water limit (and the deep-water limit, respectively). The main writing establishes the first convergence result of ILW in the periodic setting. The resolution of the deterministic convergence problem required an intricate harmonic analytic approach, particularly the Fourier restriction norm method. Moreover, the argument works well on the real line and for the analytic nonlinearity.

In the second part of this thesis, we discuss an analogous convergence result from a statistical viewpoint. More precisely, we study convergence of invariant Gibbs dynamics for ILW in the shallow-water and deep-water limits. After a brief review of the construction of the Gibbs measure for ILW, we 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. Moreover, our results hold for defocusing measures (i.e., we consider the power type nonlinearity uᵏ, for k ∈ 2N + 1).