On the blow-up scenario for some modified Serre-Green-Naghdi equations

TL;DR

This paper analyzes a modified Serre-Green-Naghdi system, establishing conditions under which solutions develop singularities in finite time, thereby advancing understanding of wave breaking phenomena.

Contribution

It provides a detailed blow-up scenario and proves finite-time singularity formation for the modified Serre-Green-Naghdi equations, extending results to include weak surface tension.

Findings

Finite-time blow-up of solutions under certain conditions

Existence of solutions developing singularities

Results applicable to systems with weak surface tension

Abstract

The present paper deals with a modified Serre-Green-Naghdi (mSGN) system that has been introduced by Clamond et al. to improve the dispersion relation. We present a precise blow-up scenario of the mSGN equations and we prove the existence of a class of solutions that develop singularities in finite time. All the presented results hold also for the Serre-Green-Naghdi system with weak surface tension.