On the well-posedness of a weakly dispersive Boussinesq system

TL;DR

This paper investigates the mathematical properties of a Boussinesq system modeling weakly nonlinear long surface waves, establishing conditions for well-posedness, ill-posedness, and singularity formation.

Contribution

It provides the first rigorous analysis of well-posedness and ill-posedness for this specific dispersive Boussinesq system, including criteria for singularity development.

Findings

Established local well-posedness under certain conditions

Identified scenarios leading to ill-posedness

Provided criteria for singularity formation

Abstract

We study the Cauchy problem for one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We establish the local well-posedness and ill-posedness of solutions to the system. We also provide criteria for the formation of singularities.