Local-in-space wave breaking criteria for a shallow water wave equation

Abstract

A shallow water wave equation including the famous Degasperis-Procesi model is considered. Firstly, the L² conservation law for the equation is derived. Secondly, using the methods of transport equation, we establish the boundedness of solutions for the shallow water wave equation. Finally, utilizing the approaches to construct Lyapunov functions, we find local-in-space wave breaking criteria of its solutions on the line $\mathbb{R}$ and circle $\mathbb{S}$ under certain restrictions on the coefficients of the equation.

Keywords:

• Local well-posedness
• shallow water wave equation
• local-in-space wave breaking criteria

Mathematics Subject Classifications:

• 35B44
• 35B45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is funded by the National Natural Science Foundation of China [grant number 12361042] and 14th Five Years' Key Discipline of Xinjiang Uygur Autonomous Region [grant number 83451378].