The local well-posedness of the coupled Ostrovsky system with low regularity

In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces H-s(R) x H-s(R) of lower order s is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in H-s(R) x H-s(R) for s > - 3/4 by using Bourgain spaces. Moreover, if s < - 3/4, it is demonstrated that one of the nonlinear iteration from the initial data to the putative solutions is discontinuous with an argument on the high-to-low frequency. In this sense, it is then concluded that the coupled Ostrovsky equations is ill-posed in H-s(R) x H-s(R) for s < - 3/4.