Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces ✩

We study the Cauchy problem of the Ostrovsky equation ∂tu−β∂3 xu−γ ∂−1 x u+u∂xu = 0, with βγ <0. By establishing a bilinear estimate on the anisotropic Bourgain space Xs,ω,b, we prove that the Cauchy problem of this equation is locally well-posed in the anisotropic Sobolev space H(s,ω)(R) for any s > −58 and some ω ∈ (0, 12 ). Using this result and conservation laws of this equation, we also prove that the Cauchy problem of this equation is globally well-posed in H(s,ω)(R) for s 0. © 2006 Elsevier Inc. All rights reserved