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.

Cahn–Hilliard方程是二元混合物中相分离的经典模型，表现出相的自发粗化。我们研究带有强对流项的Cahn–Hilliard方程，以便对相的搅拌和最终混合建模。主要结果是，如果施加的对流充分混合，则不会发生相分离，而是溶液呈指数收敛至均匀混合状态。所施加的漂移的混合效果通过相关的对流-超扩散方程的耗散时间来量化，我们给出了耗散时间较小的速度场的示例。我们还研究了该量与标准对流扩散方程的耗散时间之间的关系。