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 \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} where $u_0\in H_0^2(\mathbb T^2)$, $A,\varepsilon>0$, $a<0$, 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 $Av_1(y)\partial_xu$ models imposed stirring along the shear direction; through mixing, it enhances dissipation and counteracts the growth driven by the unstable cubic term $\Delta(au^3)$. 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 $|a|$ and $\|\int_{\mathbb T} u_0(x,\cdot)\,dx\|_{L_y^2}$ are sufficiently small.