Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces

Abstract

In this paper, we prove the solution map of the Cauchy problem of Camassa-Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood-Paley theory and the method introduced by Kato [37] and Danchin [21].

Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation

“…[11,12,15,16,41,51]. The local well-posedness for the Cauchy problem of the CH equation was investigated in [13,23,37]. Blow-up phenomena and global existence of strong solutions were discussed in [10,13,14].…”

Ill-posedness of the Camassa–Holm and related equations in the critical space

“…The analogous well-posedness result on the torus was shown in [21] (See [9] for earlier result in H 3 (T)). Danchin [13] considered the local well-posedness in the Besov space, and proved well-posedness in B s p,r if 1 ≤ p ≤ ∞, 1 ≤ r < ∞ and s > max{1 + 1/p, 3/2} (For continuous dependence, see Li and Yin [18]). Note that if one wants to include the case r = ∞ then one has to weaken the notion of well-posedness since the continuity of the solution with values in B s p,∞ as well as the continuity of the data-to-solution map with values in L ∞ (0, T ; B s p,∞ ) are not known to hold.…”

Global weak solutions for a generalized Camassa–Holm equation

“…The local well-posedness of the Cauchy problem for the CH equation in Sobolev spaces and Besov spaces was discussed in [8,9,15,16,27,35]. The global strong solutions and finite time blow-up strong solutions to the CH equation were studied in [7][8][9][10]15,16,35].…”

“…For the DP equation, the local well-posedness of the Cauchy problem in Sobolev spaces and Besov spaces was established in [24,25,27,44]. The DP equation also has global strong solutions [29,45,47] and finite time blow-up solutions [20,21,29,30,[44][45][46][47].…”