Rough solutions for the periodic Schrödinger–Korteweg–de Vries system

Abstract

We prove two new mixed sharp bilinear estimates of Schrödinger–Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger–Kortweg–de Vries (NLS–KdV) system in the periodic setting. Our lowest regularity is H 1 / 4 × L 2 , which is somewhat far from the naturally expected endpoint L 2 × 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 × H − 3 4 + . Nevertheless, we conclude the global well-posedness of the NLS–KdV system in the energy space H 1 × H 1 using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.