The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity

In this paper, we investigate the Cauchy problem for the Ostrovsky equation∂x(ut−β∂x3u+12∂x(u2))−γu=0, in the Sobolev space H−3/4(R). Here β>0(<0) corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mejía (2006) [13], (2009) [15], K. Tsugawa (2009) [26] proved that the problem is locally well-posed in Hs(R) when s>−3/4 and ill-posed when s<−3/4. By using some modified Bourgain spaces, we prove that the problem is locally well-posed in H−3/4(R) with β<0 and γ>0. The new ingredient that we introduce in this paper is Lemmas 2.1-2.6.