Refined wave breaking for the one-dimensional nonlinear shallow water equations

TL;DR

This paper provides a detailed analysis of wave breaking phenomena in the one-dimensional nonlinear shallow water equations, offering sharp lifespan estimates and constructing initial data that lead to finite-time blowup.

Contribution

It introduces a refined wave breaking description with precise lifespan estimates and constructs initial data causing finite-time blowup in solutions.

Findings

Lifespan estimates depend on amplitude and topography parameters.

Constructed initial data with finite $ ext{H}^5$-norm causes finite-time blowup.

Identified a precise blowup profile for solutions.

Abstract

This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography parameters, under a non-cavitation condition which excludes the scenario that the solutions have compact support. We construct smooth initial data with finite $\dot{H}^5$-norm such that the $L^\infty$-norm of the spatial derivative of the solution blows up at one single point in finite time with a precise blowup profile.