Local well-posedness and quantitative ill-posedness for the Ostrovsky equation

In this article we consider the initial value problem (IVP) for the Ostrovsky equation:∂tu−∂x3u∓∂x−1u+u∂xu=0,x∈R,t∈R,u(x,0)=u0(x), with initial data in Sobolev spaces Hs(R)Hs(R). We prove that if s>−34 this IVP is locally well-posed in Hs(R)Hs(R) and if s<−34 the IVP is not quantitatively well-posed in Hs(R)Hs(R).