Global controllability of the Cahn-Hilliard equation

This paper investigates the global controllability properties of the Cahn--Hilliard equation posed on the $d$-dimensional flat torus $\mathbb{T}^d$. We first establish small-time global approximate controllability of the system by means of controls acting on finitely many Fourier modes, relying on techniques inspired by geometric control theory. We then prove null controllability of the linearized equation using a spatially localized control supported on an arbitrary measurable subset of positive Lebesgue measure, based on quantitative propagation of smallness estimates for the free dynamics. For dimensions $d \in \{1,2,3\}$, we further derive local null controllability for the full nonlinear system via a fixed-point argument. By combining these results, we establish global null controllability of the Cahn--Hilliard equation. This work provides the first result on global controllability for this equation, achieved through a two-stage strategy in which the control is first localized in Fourier space and subsequently restricted to a set of positive measure.