The well-posedness for the Camassa-Holm type equations in critical Besov spaces [formula omitted] with 1 ≤ p < +∞

Abstract

For the famous Camassa-Holm equation, the well-posedness in C([0,T];Bp,11+1p(R)) with 1≤p≤2 and the ill-posedness in C([0,T];B∞,11(R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C([0,T];Bp,11+1p(R)) with 2<p<+∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C([0,T];Bp,11+1p(R)) with 1≤p<+∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness.