Long time existence for a two-dimensional strongly dispersive Boussinesq system

TL;DR

This paper establishes long-term existence of solutions for a unique two-dimensional Boussinesq system modeling water waves, overcoming linear well-posedness challenges using advanced mathematical techniques.

Contribution

It proves a long time existence result for a special 2D Boussinesq system with nontrivial zero eigenvalues, employing 'good unknowns' and normal form methods.

Findings

Long time existence of solutions demonstrated

Unique system with nontrivial zero eigenvalues addressed

Advanced techniques used to handle linear well-posedness issues

Abstract

We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the use of "good unknowns " and of normal form techniques.