In this talk, we study the effect of adding an advection term, and the resulting increased dissipation rate, on the growth of solutions to two specific non-linear parabolic PDEs. One is the well-known Cahn-Hilliard equation, and the other is used to model thin-film growth. In the classical model, the Cahn-Hilliard equation's solution spontaneously forms domains with $c=\pm 1$ separated by thin transition regions.  While, for the thin-film type equation, one can prove that the solutions starting from the initial data with negative energy will blow up in a finite time.

In contrast to the classical model, we imposed an incompressible velocity field u(t,x) to the two equations to model the stirring of the fluids. 
Our main result asserts that if the imposed velocity field is sufficiently mixing, then no phase separation occurs in the Cahn-Hilliard model, and the global existence of the solutions to the thin-film type equation can be proved. Further, both solutions will converge exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified via the associated advection-hyperdiffusion equation's dissipation time.