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

Abstract

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,&v(x,0)=ϕ(x),\\ \partial_tw + α\partial_x^3w + \partial_x(v^2w) =0,& w(x,0)=ψ(x), \end{cases} \end{equation*} and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^s(\mathbb{R})\times H^s(\mathbb{R})$, $s>-\frac{1}{2}$, for $0<\alpha <1$. Our result covers the whole scaling sub-critical range of Sobolev regularity contrary to the case $\alpha =1$, where the local well-posedness holds only for $s\ge \frac{1}{4}$. We also prove that the local well-posedness result is sharp in two different ways, viz., for $s<-\frac{1}{2}$ the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow-map that takes initial data to the solution fails to be $C^3$ at the origin. These results hold for $\alpha >1$ as well.

30 pages, to appear in Jr. of Evolution Equations - corrected version