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

摘要

We consider the initial value problem associated to a system consisting modifiedKorteweg-de Vries type equations ?tv ? 3x v ?x(vw2) = 0, v(x,0) =Φ (x),?tw α? 3x w ?x(v2w) = 0, w(x,0) =ψ(x),and using only bilinear estimates of the type Jγ F1b1·Jβ F2b2 L2xL2t, where J is the Bessel potentialand F jbj, j = 1,2 are multiplication operators, we prove the local well-posedness results forgiven data in low regularity Sobolev spaces Hs(R)×Hk(R) for α = 0,1. In this work weimprove 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 0 and k 0, in the sense of the trilinearestimates fails to hold.