Non-uniform dependence on initial data for the two-component fractional shallow water wave system. (March 2020) - Zhou, Shouming; Pan, Shihang; Mu, Chunlai; Luo, Honglin
Abstract: This paper focuses on continuity of a two-component high-order Camassa–Holm system, which was proposed by Escher and Lyons (2015). We know that its solutions depend continuously on their initial data from the local well-posedness results. In present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces H s 1 ( R ) × H s 2 ( R ) with s 1 > r + 1 2 and 1 2 < s 2 ≤ s 1 − 1 ≤ s 2 + 2 r − 2, r ∈ Z +, which improves the corresponding results for higher-order Camassa–Holm in Tang and Liu (2015) and Wang and Li (2019) to two-component and the lowest Sobolev Space.