Ill-posedness for the Camassa-Holm and related equations in Besov spaces

Abstract

In this paper, we give a construction of $u_0\in B_{p,\infty }^{\sigma }$ such that corresponding solution to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{p,\infty }^{\sigma }$, which implies the ill-posedness for this equation in $B_{p,\infty }^{\sigma }$. We also apply our method to the b-equation and Novikov equation.

arXiv comment14 pages

References in corpus (3)

• An integrable shallow water equation with peaked solitons
• 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

Cited by in corpus (2)

• The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces