The Cauchy problem of the rotation Camassa–Holm equation in equatorial water waves

ABSTRACT

In this paper, we first establish the local well-posedness and the rotational speed limit for the rotation Camassa–Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces $B_{p,r}^s$ with $s>max(1+\frac{1}{p},\frac{3}{2})$ or $s=1+\frac{1}{p},r=1$. Then, we prove the local ill-posedness of the rotation Camassa–Holm equation in $B_{p,r}^{1+\frac{1}{p}},p\in [1,+\mathrm{\infty }],r\in (1,+\mathrm{\infty }]$. Finally, we show the existence of global weak solutions for the rotation Camassa–Holm equation.

COMMUNICATED BY:

• Adrian Constantin

Keywords:

• The rotation Camassa–Holm equation
• local well-posedness
• the rotational speed limit
• local ill-posedness
• global weak solutions

Mathematics Subject Classifications:

• 35A01
• 35Q53
• 35B30
• 35D30
• 35G25

Acknowledgements

The authors thank the referees for valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by National Natural Science Foundation of China – NNSFC [No. 11671407], Fundo para o Desenvolvimento das Ciências e da Tecnologia – FDCT [No. 0091/2013/A3], Guangdong Special Support Program [No. 8-2015] and the key project of NSF of Guangdong Province [No. 2016A030311004].