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The Cauchy problem for nonlinear dispersive models of long internal waves in the presence of the Coriolis force

Ricardo Freire, Thyago S. R. Santos

TL;DR

The paper analyzes nonlinear, nonlocal dispersive models for long internal waves in a rotating frame, focusing on the Rotational Modified Benjamin-Ono and Rotational Modified ILW equations with a Coriolis term $\gamma\partial_x^{-1}u$. It develops an unconditional well-posedness theory in the low-regularity spaces $Z^s_x(\mathbb{R})$ for $s>\tfrac12$ using a dyadic, energy-based method inspired by Molinet–Vento, and establishes global well-posedness for $s\ge 1$ via conserved quantities. An ill-posedness result rules out $C^2$-smooth data-to-solution maps in these spaces, highlighting the delicate balance between nonlocal dispersion and rotation. The analysis employs Bourgain spaces $X^{s,b}$, a detailed Littlewood–Paley decomposition, and pseudoproduct techniques to control resonant interactions, with extensions to the ILW class through the Dirichlet-to-Neumann type operator $\mathcal{T}_\delta$ and discussion of the deep-water limit. The results illuminate how Coriolis effects interact with nonlocal dispersion in a rigorous well-posedness framework and lay groundwork for broader dispersion perturbations $D^\alpha_x$ and related rotating-fluid models.

Abstract

We investigate models of dispersive long internal waves with rotational effects, specifically the Benjamin-Ono (BO) and intermediate long wave (ILW) equations modified by the presence of the nonlocal operator $\partial_x^{-1}$, which mathematically accounts for rotational influences. We establish a local and global well-posedness theory while ensuring the unconditional uniqueness of solutions in low-regularity Sobolev-type spaces. Our approach is based on techniques introduced by Molinet and Vento in \cite{MR3397003}.

The Cauchy problem for nonlinear dispersive models of long internal waves in the presence of the Coriolis force

TL;DR

The paper analyzes nonlinear, nonlocal dispersive models for long internal waves in a rotating frame, focusing on the Rotational Modified Benjamin-Ono and Rotational Modified ILW equations with a Coriolis term . It develops an unconditional well-posedness theory in the low-regularity spaces for using a dyadic, energy-based method inspired by Molinet–Vento, and establishes global well-posedness for via conserved quantities. An ill-posedness result rules out -smooth data-to-solution maps in these spaces, highlighting the delicate balance between nonlocal dispersion and rotation. The analysis employs Bourgain spaces , a detailed Littlewood–Paley decomposition, and pseudoproduct techniques to control resonant interactions, with extensions to the ILW class through the Dirichlet-to-Neumann type operator and discussion of the deep-water limit. The results illuminate how Coriolis effects interact with nonlocal dispersion in a rigorous well-posedness framework and lay groundwork for broader dispersion perturbations and related rotating-fluid models.

Abstract

We investigate models of dispersive long internal waves with rotational effects, specifically the Benjamin-Ono (BO) and intermediate long wave (ILW) equations modified by the presence of the nonlocal operator , which mathematically accounts for rotational influences. We establish a local and global well-posedness theory while ensuring the unconditional uniqueness of solutions in low-regularity Sobolev-type spaces. Our approach is based on techniques introduced by Molinet and Vento in \cite{MR3397003}.

Paper Structure

This paper contains 12 sections, 18 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Notations and basic concepts
  3. Preliminary results
  4. Ill-posedness result
  5. The Rotational Modified Benjamin-Ono Equation
  6. A Priori Estimates
  7. Local Theory
  8. Continuity of the Data-Solution Map
  9. Global Theory
  10. The Rotational Modified Intermediate Long Wave Equation
  11. Local Theory
  12. Global Theory

Key Result

Theorem 1.1

Let $s \in \mathbb{R}$ and $T>0$ . Then there does not exist a space $X_T$ continuously embedded in $C([-T, T], Z^s_x(\mathbb{R}))$ such that and where $\{V(t)\}_{t \in \mathbb{R}}$ the free evolution group associated to equation BO2.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Local well-posedness for \ref{['BO2']}
  • Remark 1.4
  • Theorem 1.5: Global well-posedness for \ref{['BO2']}
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Local well-posedness for \ref{['ILW2']}
  • Theorem 1.9: Global well-posedness for \ref{['ILW2']}
  • Theorem 2.1: See Muscalu_Schlag_2013
  • ...and 12 more