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 \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}.

References in corpus (5)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
• A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations
• Modeling shallow water waves
• Hydrodynamic models and confinement effects by horizontal boundaries

Cited by in corpus (3)

• Well-posedness and singularity formation for inviscid Keller-Segel-fluid system of consumption type
• Finite time blow-up for some parabolic systems arising in turbulence theory
• Global weak solutions of the Serre-Green-Naghdi equations with surface tension