Rough solutions for the periodic Schrodinger-Korteweg-de Vries system

We prove two new mixed sharp bilinear estimates of Schrodinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrodinger-Kortweg-de Vries (NLS-KdV) system in the periodic setting. Our lowest regularity is H-1/4 x L-2, which is somewhat far from the naturally expected endpoint L-2 x H-1/2. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint L-2 x H-3/4+. Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space H-1 x H-1 using our local well-posedness result and three conservation laws discovered by M. Tsutsumi. (c) 2006 Elsevier Inc. All rights reserved.