A remark on the local well-posedness for a coupled system of mKdV type equations in H^s × H^k

Abstract

. We consider the initial value problem associated to a system consisting modi fi ed Korteweg-de Vries type equations and using only bilinear estimates of the type (cid:2) J γ F 1 b 1 J F 2 b 2 (cid:2) L 2 x L 2 t , where J is the Bessel potential and F jb j , j = 1 , 2 are multiplication operators, we prove the local well-posedness results for given data in low regularity Sobolev spaces H s ( R ) × H k ( R ) for α (cid:3) = 0 , 1. In this work we improve the previous result in [6], extending the LWP region from | s − k | < 1 / 2 to | s − k | < 1. This result is sharp in the region of the LWP with s (cid:2) 0 and k (cid:2) 0, in the sense of the trilinear estimates fails to hold.