Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems

TL;DR

This paper proves finite-time singularity formation in solutions to the Serre-Green-Naghdi and abcd-Boussinesq equations, revealing conditions under which solutions break down or change sign, with implications for water wave modeling.

Contribution

It establishes new finite-time singularity results for the Serre-Green-Naghdi and abcd-Boussinesq systems, extending understanding of their solution behaviors.

Findings

Solutions to the Serre-Green-Naghdi equation cannot be globally defined when reaching the bottom tangentially.

Solutions to the abcd-Boussinesq system can change sign in finite time.

Finite time singularity scenarios are proved for the abcd-Boussinesq system.

Abstract

In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper \emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.} Furthermore, we also prove that the solution to the $abcd-$Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for the $abcd-$Boussinesq system. These latter mathematical results are related to the numerics in \emph{Bona, \& Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016}.