Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces

Abstract

In the paper, by constructing a initial data $u_0\in B_{p,\infty }^{\sigma }$ with $\sigma -2>\max \{1+\frac{1}{p},\frac{3}{2}\}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_0$ is discontinuous at $t=0$ in the norm of $B_{p,\infty }^{\sigma }$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B_{p,\infty }^{\sigma }$.

arXiv comment13 pages. arXiv admin note: text overlap with arXiv:2104.05973 by other authors

References in corpus (8)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs
• Ill-posedness of the Camassa-Holm and related equations in the critical space
• On the Euler-Poincaré equation with non-zero dispersion
• Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincaré equations
• Ill-posedness for the Camassa-Holm and related equations in Besov spaces
• A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B_{p,1}^{1+\frac{1}{p}},1\le p<+\infty$