Non-uniform dependence for the periodic higher dimensional Camassa–Holm equations

Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system

“…Lately, Li and Yin [29] proved that the corresponding solution is continuous dependence for the initial data in Besov spaces. Zhao, Yang and Li [34] showed that the solution map of the periodic Euler-Poincaré system is not uniformly continuous in Besov space B s 2,r , s > 1 + d 2 . In this paper, inspired by [25,26], we will show that the solution map of (1.1) is not uniformly continuous dependence in Sobolev space H s (R d ).…”

On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces

“…Later, Luo and Yin have obtained a new blow-up result in the periodic case by using the rotational invariant properties of the equation [25]. For more results of Euler-Poincaré equations, see [25,32].…”

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

“…Also, Li, Dai and Li in [38] have shown that the data-to-solution map for (1.1) is not uniformly continuous dependence in Besov spaces B s p,r (R d ), s > max{1 + d 2 , 3 2 }. For more results of higher dimensional Camassa-Holm equations, see [36,46].…”