Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow

In this paper, we study the advective unstable Cahn–Hilliard equation on ${\mathbb{T}}^2$ with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon \Delta^2 u= \Delta(a u^3+ b u^2) \quad & \quad $\text{on}$ \quad \mathbb T^2; \ \ u \ $\text{periodic}$ \quad & \quad $\text{on}$ \quad \partial \mathbb T^2, \end{cases} \end{equation*} where $u_0\in H_0^2({\mathbb{T}}^2)$, $A,\epsilon >0$, $a<0$, and $b\in \mathbb{R}$. The condition $a<0$ puts the model in an unstable phase-field regime: the nonlinear chemical potential may amplify, rather than restore, concentration fluctuations, as in spinodal decomposition. The shear term $A{v}_1(y){\mathrm{\partial }}_{x}u$ models imposed stirring along the shear direction; through mixing, it enhances dissipation and counteracts the growth driven by the unstable cubic term $\mathrm{\Delta }(a{u}^3)$. Assuming that the shear profile has finitely many critical points and that linearly growing modes occur only in the shear direction, we prove that the $L^2$-energy converges exponentially to zero, provided $\mathrm{\mid }a\mathrm{\mid }$ and $\mathrm{\parallel }{\int }_{\mathbb{T}}{u}_0(x,\cdot )dx{\mathrm{\parallel }}_{L_y^2}$ are sufficiently small.