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

Abstract

In this paper, we investigate the continuous dependence on initial data of solutions to the Euler-Poincaré system. By constructing a sequence approximate solutions and calculating the error terms, we show that the data-tosolution map is not uniformly continuous in Sobolev space H s (R d ) for s > 1 + d 2 .2010 Mathematics Subject Classification. 35Q35.

Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces

“…Lately, Li and Yin [33] proved that the corresponding solution is continuous dependence for the initial data in Besov spaces. Inspired by [25,26], Li, Dai and Zhu [34] show that the corresponding solution is not uniformly continuous dependence for the initial data in Sobolev spaces…”

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

“…Most previous work on constinuity has focused on the spacial one-dimensional Camassa-Holm type equations equations, for the multi-dimensional Euler-Poincaré equations (1.1), the continuity problem has not been thoroughly investigated. Until recently, Li et al [24] shown that the corresponding solution to (1.1) is not uniformly constinuous dependence for that the initial data in…”

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

“…Taking advantage of the Littlewood-Paley decomposition theory, Yan and Yin [44] further discussed the local existence and uniqueness of the solution to (1.1) in Besov spaces B s p,r (R d ) with s > max{1 + d p , 3 2 } and s = 1 + d p , 1 ≤ p ≤ 2d, r = 1. Recently, Li, Dai and Zhu [34] shown that the corresponding solution to (1.1) is not uniformly constinuous dependence for that the initial data in H s (R d ), s > 1 + d 2 . 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 }.…”