Ill_posedness for a two_component Novikov system in Besov space

Abstract: In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data $(\rho_0, u_0)$ in $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ with $s>\max{2+\frac{1}{p}, \frac{5}{2}}$ and $1\leq p \leq \infty$, we show that any energy bounded solution starting from $(\rho_0, u_0)$ does not converge back to $(\rho_0, u_0)$ in the metric of $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.