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 partial derivative x (u(t) - beta partial derivative(3)(x)u + 1/2 partial derivative x(u(2))) - gamma u = 0, in the Sobolev space H-3/4(R). Here beta > 0(< 0) corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mejia (2006) [13], (2009) [15], K. Tsugawa (2009) [26] proved that the problem is locally well-posed in H-s (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 beta < 0 and gamma > 0. The new ingredient that we introduce in this paper is Lemmas 2.1-2.6. (C) 2015 Elsevier Inc. All rights reserved.