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

TL;DR

This paper establishes sharp local well-posedness results for a coupled mKdV system in low regularity Sobolev spaces, extending known results to a broader range of parameters and demonstrating the limits of well-posedness.

Contribution

It proves local well-posedness for the coupled mKdV system in Sobolev spaces with regularity above -1/2 for all 0<α<1, and shows this result is optimal.

Findings

Well-posedness holds for s > -1/2, covering the entire sub-critical range.

Failure of key estimates and flow-map regularity below s = -1/2.

Results apply for α > 1 as well.

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)=\phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) =0,& w(x,0)=\psi(x), \end{cases} \end{equation*} and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$, $s> -\frac12$, 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 $s\geq \frac14$. We also prove that the local well-posedness result is sharp in two different ways, viz., for $s<-\frac12$ 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.