  We consider following fourth-order parabolic equation with gradient
nonlinearity on the two-dimensional torus with and without advection of an
incompressible vector field in the case $2<p<3$: \begin{equation*}
  \partial_t u + (-\Delta)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u).
\end{equation*} The study of this form of equations arises from mathematical
models that simulate the epitaxial growth of the thin film. We prove the local
existence of mild solutions for any initial data lies in $L^2$ in both cases.
Our main result is: in the advective case, if the imposed advection is
sufficiently mixing, then the global existence of solution can be proved, and
the solution will converge exponentially to a homogeneous mixed state. While in
the absence of advection, there exist initial data in $H^2\cap W^{1,\infty}$
such that the solution will blow up in finite time.
