The Cauchy problem for quadratic and cubic Ostrovsky equation with negative dispersion

In this paper, we consider the Cauchy problem for the quadratic and cubic Ostrovsky equation with negative dispersion ∂xut−β∂x3u+1k∂x(uk)−γu=0,β<0,γ>0,(k=2,3).Firstly, by using the Strichartz estimates instead of the Cauchy-Schwarz inequalities, we give an alternative proof of Lemma 1.2 of Isaza and Mejía (2006). Secondly, by using the Strichartz estimates instead of the Cauchy-Schwarz inequalities, we give an alternative proof of Lemma 1.3 of Isaza and Mejía (2007). Thirdly, we prove that the Cauchy problem for the cubic Ostrovsky equation is locally well-posed in Hs(R) with s≥14. Finally, we prove that the Cauchy problem for the cubic Ostrovsky equation is not well-posed in Hs(R) with s<14.