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 modified Korteweg-de Vries type equationsand using only bilinear estimates of the typewhere J is the Bessel potential and F j b 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 = 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 0 and k 0 , in the sense of the trilinear estimates fails to hold.

Locations

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