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

Abstract

In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces $H^s\left(R\right)\times H^s\left(R\right)$ of lower order $s$ is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in $H^s\left(R\right)\times H^s\left(R\right)$ for $s>-\frac{3}{4}$ by using Bourgain spaces. Moreover, if $s<-\frac{3}{4}$, 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 $H^s\left(R\right)\times H^s\left(R\right)$ for $s<-\frac{3}{4}$.