Local well-posedness for a system of modified KdV equations in modulation spaces

TL;DR

This paper establishes local well-posedness for a system of modified KdV equations in modulation spaces, addressing challenges from distinct Fourier supports and resonant interactions.

Contribution

It introduces new trilinear estimates in modulation spaces for the mKdV system with different Fourier support curves, proving well-posedness under specific regularity conditions.

Findings

Proves local well-posedness for initial data in $M_s^{2,p}$ with $s>1/4 - 1/p$

Develops novel trilinear estimates considering distinct cubic Fourier supports

Handles resonant interactions with more restrictive regularity conditions

Abstract

In this work, we consider the initial value problem (IVP) for a system of modified Korteweg-de Vries (mKdV) equations \begin{equation} \begin{cases} \partial_t v + \partial_x^3 v+ \partial_x (v w^2) = 0, \hspace{0.98 cm} v(x,0)=\psi(x),\\ \partial_t w + \alpha \partial_x^3 w+\partial_x (v^2 w) = 0,\hspace{0.5 cm} w(x,0)=\phi(x). \end{cases} \end{equation} The main interest is in addressing the well-posedness issues of the IVP when the initial data are considered in the modulation space $M_s^{2,p}(\mathbb{R})$, $p\geq 2$. In the case when $0<\alpha\ne 1$, we derive new trilinear estimates in these spaces and prove that the IVP is locally well-posed for data in $M_s^{2,p}(\mathbb{R})$ whenever $s> \frac14-\frac{1}{p}$ and $p\geq 2$. In deriving the trilinear estimate, the fact that the Fourier supports of the solution components $v$ and $w$ lie on distinct cubic curves, namely $\tau = \xi^3$ and $\tau = \alpha\xi^3$, introduces additional difficulties in handling the resonant case. This makes the analysis substantially different from what one encounters in the single-equation setting. To overcome the difficulties arising in the resonant case, it was necessary to impose the more restrictive condition $s> \frac14-\frac{1}{p}$ on the trilinear estimate, rather than the natural threshold $s> \frac14-\frac{3}{2p}$ , which would otherwise yield sharp local well-posedness for $s>-\frac12$ when $p=2$.