The Cauchy problem for a two-component Novikov equation in the critical Besov space

In this paper, we consider the Cauchy problem for a two-component Novikov equation in the critical Besov space B2,15/2. We first derive a new a priori   estimate for the 1-D transport equation in B2,∞3/2, which is the endpoint case. Then we apply this a priori   estimate and the Osgood lemma to prove the local existence. Moreover, we also show that the solution map u0↦uu0↦u is Hölder continuous in B2,15/2 equipped with weaker topology. It is worth mentioning that our method is different from the previous one that involves extracting a convergent subsequence from an iterative sequence in critical Besov spaces.