Local and global well-posedness for the Ostrovsky equation

We consider the initial value problem for (0.1) delta(t)u - beta delta(3)(x)u - gamma delta(-1)(x)u + uu(x) = 0, x,t is an element of R where u is a real valued function, beta and gamma are real numbers such that beta center dot gamma not equal 0 and delta(-1)(x) f = ((i xi)(-1) (f) over cap (xi))(V). This equation differs from Korteweg-de Vries equation in a nonlocal term. Nevertheless, we obtained local well-posedness in X-s = {f is an element of Hs (R) : delta(-1)(x) f is an element of L-2 (R)}, s > 3/4 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 beta center dot gamma > 0, we also obtain a global result in X-1, using appropriate conservation laws. (c) 2005 Elsevier Inc. All rights reserved.