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

الأصل · EN

In this paper, we investigate the Cauchy problem for the Ostrovsky equation eqnarray* ∂ₓ(uₜ-β∂ₓ³u +1/2∂ₓ(u²)) -γu=0, eqnarray* in the Sobolev space H⁻³/⁴(). Here β>0(<0) corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mejıa (J. Diff. Eqns. 230(2006), 601-681; Nonli. Anal. 70(2009), 2306-2316), K. Tsugawa (J. Diff. Eqns. 247(2009), 3163-3180) proved that the problem is locally well-posed in Hˢ() 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() with β<0 and γ>0. The new ingredient that we introduce in this paper is Lemmas 2.1-2.6.