Small data global solutions for the Camassa-Choi equations

TL;DR

This paper proves the existence and long-term behavior of small, smooth, localized solutions to a generalized internal-wave model extending the Benjamin-Ono and Intermediate Long Wave equations, relevant for fluid dynamics.

Contribution

It establishes global solutions and analyzes their long-time dynamics for a generalized Camassa-Choi internal-wave model with weak transverse effects.

Findings

Proved existence of global solutions for small initial data.

Analyzed long-time dynamics of solutions.

Extended results to a generalized internal-wave model.

Abstract

We consider solutions to the Cauchy problem for an internal-wave model derived by Camassa-Choi in a paper in Journal of Fluid Mechanics (1996). This model is a natural generalization of the Benjamin-Ono and Intermediate Long Wave equations in the case of weak transverse effects. We prove the existence and long-time dynamics of global solutions from small, smooth, spatially localized initial data on $\mathbb{R}^2$.