Hölder continuity of the solution map for the Novikov equation

Abstract

The Novikov equation (NE) has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the Camassa-Holm and Degasperis-Procesi equations, which have quadratic nonlinearities. NE is well-posed in Sobolev spaces Hs on both the line and the circle for s > 3/2, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. This work studies the continuity properties of NE further. For initial data in Hs, s > 3/2, it is shown that the solution map for NE is Hölder continuous in Hr-topology for all 0 ⩽ r < s with exponent α depending on s and r.

Well-posedness and peakons for a higher-orderμ-Camassa–Holm equation

“…(see [27]) If r > 1 2 , then there exists a constant c r > 0 depending only on r such that Proof. The proof can be done by adapting analogous methods as in [26], in which they only considered the case j = 1. For the reader's convenience, we provide the arguments with obvious modifications.…”

Continuity properties of the data-to-solution map for the two-component higher order Camassa–Holm system

“…For the reader's convenience, we provide the arguments with obvious modifications. Similar as the proof of Lemma 3 on R in [23], we can obtain…”

“…The present paper is devoted to establishing the Hölder continuity of the data-to-solution map for system (1.1) with σ = 2 in H s (R) × H s−2 (R), s > 7 2 , which provides more information about the stability of the solution map than the one given by Corollary 3.1.2 in [37]. We mention that Hölder continuity for the b-equation was proved on the line by Chen, Liu and Zhang in [5], and for other equations were showed in [23,26,31,35]. To obtain the desired result, we need to extend the estimate of f g H r−1 (R) for 0 ≤ r ≤ 1 in [23], commonly used in the previous works, to that of f g H r−k (R) for 0 ≤ r ≤ k and k > 1, which plays a key role in proving the main result.…”

“…We mention that Hölder continuity for the b-equation was proved on the line by Chen, Liu and Zhang in [5], and for other equations were showed in [23,26,31,35]. To obtain the desired result, we need to extend the estimate of f g H r−1 (R) for 0 ≤ r ≤ 1 in [23], commonly used in the previous works, to that of f g H r−k (R) for 0 ≤ r ≤ k and k > 1, which plays a key role in proving the main result.…”

“…Proof. The proof can be done by adapting analogous methods as in [23], in which they only considered the case k = 1. For the reader's convenience, we provide the arguments with obvious modifications.…”