Global Existence for the two-dimensional Cahn-Hilliard equation with a Shear Flow

In this paper, we consider the advective Cahn-Hilliard equation in 2D with shear flow: $$ \begin{cases} u_t+v_1(y) \partial_x u+\gamma \Delta^2 u=\gamma \Delta(u^3-u) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} $$ where $\mathbb T^2$ is the two-dimensional torus. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the global existence of solutions with arbitrary initial $H^2$ data. The main difficulty of this paper is to handle the high-regularity and non-linearity underlying the term $\Delta(u^3)$ in a proper way. For such a purpose, we modify the methods by Iyer, Xu, and Zlatos in 2021 under a shear flow setting.