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

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 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 Bona, and Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016.