Non-uniform Dependence on Initial Data for the Camassa–Holm Equation in the Critical Besov Space

Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces

“…We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) [20,21] on Camassa-Holm equation with Q = 1 and extends the previous non-uniform continuity in Sobolev spaces (2015) [23] to Besov spaces. In addition, the non-uniform continuity in critical space B 3 2 2,1 (R) is the first to be considered in our paper.…”

Ill-posedness for the Camassa-Holm equation in $B_{p,1}^{1}\cap C^{0,1}$

“…2,1 in [24,25] (see [19,20,21] for earlier results in H s ). Danchin [15] obtained the ill-posedness of (1.1) in B 3/2 2,∞ (the data-to-solution map is not continuous by using peakon solution).…”

Non-uniform dependence for the Camassa-Holm and Novikov equations in low regularity Besov spaces

“…Himonas-Misiołek [21] obtained the first result on the non-uniform dependence for (1.1) in H s (T) with s ≥ 2 using explicitly constructed travelling wave solutions, which was sharpened to s > 3 2 by Himonas-Kenig [19] on the real-line and Himonas-Kenig-Misiołek [20] on the circle. In our recent papers [25,26], we proved the non-uniform dependence on initial data for (1.1) under both the framework of Besov spaces B s p,r for s > max 1…”

“…Remark 1.1. Compared with [25,26], Theorem 1.1 is new since the regularity index is enlarged to s > 1. In this sense, Theorem 1.1 improves the previous results in [25,26].…”

“…Another important integrable equation similar to the CH equation (1.1) is the Novikov equation (see, e.g., [17,18,24,25])…”