  The Cahn--Hilliard equation is a classic model of phase separation in binary
mixtures that exhibits spontaneous coarsening of the phases. We study the
Cahn--Hilliard equation with an imposed advection term in order to model the
stirring and eventual mixing of the phases. The main result is that if the
imposed advection is sufficiently mixing then no phase separation occurs, and
the solution instead converges exponentially to a homogeneous mixed state. The
mixing effectiveness of the imposed drift is quantified in terms of the
dissipation time of the associated advection-hyperdiffusion equation, and we
produce examples of velocity fields with a small dissipation time. We also
study the relationship between this quantity and the dissipation time of the
standard advection-diffusion equation.

Analysis of PDEs;Chaotic Dynamics;Fluid Dynamics