Local and global well-posedness for the Ostrovsky equation

We consider the initial value problem forequation(0.1)∂tu-β∂x3u-γ∂x-1u+uux=0,x,t∈R,where u   is a real valued function, ββ and γγ are real numbers such that β·γ≠0β·γ≠0 and ∂x-1f=((iξ)-1f^(ξ))∨.This equation differs from Korteweg–de Vries equation in a nonlocal term. Nevertheless, we obtained local well-posedness in Xs={f∈Hs(R):∂x-1f∈L2(R)},s>34, using techniques developed in [C.E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg–de Vries equation, J. Amer. Math. Soc. 4 (1991) 323–347]. For the case β·γ>0β·γ>0, we also obtain a global result in X1X1, using appropriate conservation laws.