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

Abstract

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: \[ \left\{\begin{aligned} & u_t + u_{xxx} + vv_x = 0 , \\ & v_t + \alpha v_{xxx} + (uv)_x = 0 , \\ & (u,v) \mid_{t=0} = (u_0, v_0) \in H^s(\mathbb{T}) \times H^s(\mathbb{T}), \end{aligned}\right. \qquad x \in \mathbb{T}, \, t\in \mathbb{R}, \] 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 $s^{*}(\alpha)$ of the Sobolev regularity of the initial data for $C^3$ local well-posedness. In particular, Oh showed that \[ s^{*}(\alpha) = \left\{ \begin{array}{lll} 1, & \text{for $\alpha$ such that $\sqrt{12/\alpha - 3} \in \mathbb{Q}$ }, \\ \frac12, & \text{for a.e. $\alpha$ such that $\sqrt{12/\alpha - 3} \notin \mathbb{Q}$ }. \end{array}\right. \] In this paper, by allowing the mean of $u_0$ to be non-zero, we find that the critical index $s^{*}(\alpha)$ can be lowered from $1$ to $\frac12$ when $\sqrt{12/\alpha - 3} \in \mathbb{Q}$. For other values of $\alpha$, except in a set of zero measure, we also justify the critical index $s^{*}(\alpha)$ to be $\frac12$ 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.