Well-posedness of the modified Camassa–Holm equation in Besov spaces
In this paper, we consider the modified Camassa–Holm equation of the form 

$$y_t + 2 u_x y + uy_x = 0, \quad y = (1 - \partial_x^2)^{2}u.$$

We prove that the Cauchy problem for this equation is locally well-posed in the critical Besov space \({B_{2, 1}^{7/2}}\) or in \({B_{p, r}^{s}}\) with \({1\leq p, r\leq + \infty}\), \({s > \max\{3 + 1/p, 7/2\}}\). Particularly, our method used to prove the local well-posedness in \({B_{2, 1}^{7/2}}\) is different from the previous one used in critical Besov space which involves extracting a convergent subsequence from an iterative sequence. We also prove that if a weaker \({B_{p, r}^q}\)-topology is used, then the solution map becomes Holder continuous. Furthermore, we obtain the peakon-like solution which enable us to prove the ill-posedness in \({B_{2, \infty}^{7/2}}\). Finally, when \({x \in \mathbb{T} = \mathbb{R}/2 \pi \mathbb{Z}}\), we show that the solution map is not uniformly continuous in \({B_{2, r}^{s}}\) with \({1\leq r\leq \infty}\) and \({s > 7/2}\) or \({r = 1, s = 7/2}\).