In this paper we mainly investigate the Cauchy problem of a two-component Novikov system. We first prove the local well-posedness of the system in Besov spaces $B^{ s-1}_{ p,r}\times B^s_{ p,r}$ with $p,r\in[1,\infty],~s>\max\{ 1+\frac{ 1}{ p},\frac{ 3}{ 2}\}$ by using the Littlewood-Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we establish the local well-posedness of the system in the critical Besov space $B^{ \frac{ 1}{ 2}}_{ 2,1}\times B^{ \frac{ 3}{ 2}}_{ 2,1}$. Moreover, we present two blow-up criteria for the system by making use of the conservation laws.
Categories: math.AP