Sharp well-posedness for a coupled system of mKdV type equations

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $${\mathrm{\partial }}_{t}v+{\mathrm{\partial }}_x^{3}v+{\mathrm{\partial }}_x(v{w}^2)=0,v(x,0)=\varphi (x),$$ $${\mathrm{\partial }}_{t}w+\alpha {\mathrm{\partial }}_x^{3}w+{\mathrm{\partial }}_x(v^{2}w)=0,w(x,0)=\psi (x),$$ and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}($\text{I}$!$\text{R}$)\times H^{k}($\text{I}$!$\text{R}$)$,$s,k> -\frac12$and$|s-k|\leq 1/2$,for$\alpha\neq 0,1$. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $C^3$ at the origin, when $s<-1\mathrm{/}2$ or $k<-1\mathrm{/}2$ or $\mathrm{\mid }s-k\mathrm{\mid }>2$; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1\mathrm{/}2$ (b) $k-2s>1$ or $s<-1\mathrm{/}2$; © $s=k=-1\mathrm{/}2$; (III) the local well-posedness result is sharp in a sense that we can not reduce the proof of the trilinear estimates, proving some related bilinear estimates (as in Tao [19]).