  In this paper, we consider the advective unstable Cahn-Hilliard equation in
2D 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*} with an initial data $u_0 \in H_0^2(\mathbb T^2)$, where
$\mathbb T^2$ is the two-dimensional torus, $A, \varepsilon>0$, $a<0$, $b \in
\mathbb R$. 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 $L^2$-energy of the solutions to such problems converges
expotentially to zero, if in addition, both $|a|$ and $\left\| \int_{\mathbb T}
u_0(x, \cdot ) dx \right\|_{L_y^2}$ are sufficiently small.
