Global well-posedness of the periodic Majda-Biello systems in the resonant case

We study the Cauchy problem for the following Majda-Biello system in the resonant case $\alpha \in (0,4]\setminus \{1\}$ on the periodic domain $\mathbb{T}$. $$\{\begin{array}{rclllll}{u}_t+u_{xxx}& =& -v{v}_x,v_t+\alpha v_{xxx}& =& -(uv{)}_x,(u,v){\mathrm{\mid }}_{t=0}& =& (u_0,v_0),\end{array}x\in \mathbb{T},t\in \mathbb{R}\mathrm{.}$$ Here, the initial data $u_0$ and $v_0$ live in the Sobolev space $H^s(\mathbb{T})$ and $u_0$ has zero mean over $\mathbb{T}$. We establish the optimal global well-posedness for the above system when $s\ge \min \{1,s_{\alpha }+\}$, where $s_{\alpha }$ is a constant depending only on $\alpha$, and $s_{\alpha }=\frac{1}{2}$ for almost every $\alpha \in (0,4]\setminus \{1\}$. These indices match the sharp ones in the local theory obtained by Oh (Int. Math. Res. Not., 18:3516-3556, 2009). The main novelty of our proof is to introduce a third modified energy in the I-method by performing a partial correction to the second modified energy around the resonant region. This partial correction avoids the singularity in the full correction, and therefore makes it possible to pointwise bound the quintilinear multipliers arising in the time derivative of the third modified energy. Another feature in our proof is the combination of the Diophantine approximation theory with the sparsity of nearly resonant frequency pairs on the periodic lattice to obtain the key quadrilinear and quintilinear estimates, leading to an almost conservation law sufficient for the global iteration.