Mean Effects on Critical Well-Posedness for Majda-Biello Systems on the Torus

This paper studies how the mean of the initial data $u_0$ affects the critical indices concerning local well-posedness for the following Majda-Biello systems: $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle u_t+u_{xxx}+v{v}_x=0,}\\ {\displaystyle }& {\displaystyle v_t+\alpha v_{xxx}+(uv{)}_x=0,}\\ {\displaystyle }& {\displaystyle (u,v){\mid }_{t=0}=(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T}),}\end{array}x\in \mathbb{T},t\in \mathbb{R},$$ where $\mathbb{T}$ refers to the periodic torus and the dispersion coefficient $\alpha$ is restricted in $(0,4]\setminus \{1\}$ which corresponds to resonant cases. Previously, under the zero-mean assumption on $u_0$, Oh (Int. Math. Res. Not., (18):3516-3556, 2009) determined the critical indices $s^{\ast }(\alpha )$ of the Sobolev regularity of the initial data for $C^3$ local well-posedness. In particular, Oh showed that $$s^{\ast }(\alpha )=\{\begin{array}{ll}1,& \text{for }\alpha \text{ such that }\sqrt{12\mathrm{/}\alpha -3}\in \mathbb{Q},\\ \frac{1}{2},& \text{for a.e. }\alpha \text{ such that }\sqrt{12\mathrm{/}\alpha -3}\mathrm{\notin }\mathbb{Q}\mathrm{.}\end{array}$$ In this paper, by allowing the mean of $u_0$ to be non-zero, we find that the critical index $s^{\ast }(\alpha )$ can be lowered from $1$ to $\frac{1}{2}$ when $\sqrt{12\mathrm{/}\alpha -3}\in \mathbb{Q}$. For other values of $\alpha$, except in a set of zero measure, we also justify the critical index $s^{\ast }(\alpha )$ to be $\frac{1}{2}$ regardless of the mean of $u_0$. By subtracting the mean from $u_0$, the original Majda-Biello systems are slightly modified to contain first-order terms but with zero-mean initial data. The key ingredient in our proof is to introduce a refined Diophantine approximation theory to capture the essential resonance effect for the perturbed dispersive structure caused by these additional first-order terms.