Long time existence for a class of weakly transverse Boussinesq systems

TL;DR

This paper establishes long-time existence results for solutions to a complex, anisotropic Boussinesq system modeling surface water waves, overcoming derivative loss through symmetrization techniques.

Contribution

It introduces a novel symmetrization approach for anisotropic Boussinesq systems with directional dispersion, enabling long-time existence proofs.

Findings

Proves long-time existence of solutions for the weakly transverse Boussinesq system.

Handles derivative loss due to anisotropic dispersion by symmetrization.

Extends understanding of water wave models with directional dispersion.

Abstract

We prove the existence on long time scales of the solutions to the Cauchy problem for a version of weakly transverse Boussinesq systems arising in the modeling of surface water waves. This system is much more complicated than the isotropic Boussinesq systems because dispersion is only present in the x-direction, leading to anisotropic eigenvalues in the linearized system. This anisotropic character leads to loss of y-derivatives for the solutions. To overcome this main difficulty our strategy is to symmetrize the system by introducing suitable good unknowns in the sense of [3].