ON LOCAL WELL-POSEDNESS AND ILL-POSEDNESS RESULTS FOR A COUPLED SYSTEM OF MKDV TYPE EQUATIONS

摘要

We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations ｛∂_tv + ∂_x~3v + ∂_x(vw~2) = 0, v(x,0) = Φ(ⅹ), ∂_tw + α∂_x~3w + ∂_x(v~2w) = 0, w(x,0) = ψ(ⅹ), and prove the local well-posedness results for a given data in low regularity Sobolev spaces H~S(IR) × H~k(IR), s,k ＞ -1/2 and |s - k| ＜ 1/2, for α≠ 0,1. Also, we prove that: (Ⅰ) the solution mapping that takes initial data to the solution fails to be C~3 at the origin, when s ＜-1/2 or k ＜-1/2 or 丨s- k丨 ＞ 2; (Ⅱ) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) s - 2k ＞ 1 or k ＜ -1/2 (b) k - 2s ＞ 1 or s ＜ -1/2; (c) s = k = -1/2.