Suppression of epitaxial thin film growth by mixing

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.