On the Cahn–Hilliard equation with no-flux and strong anchoring conditions

Abstract

The Cahn–Hilliard equation is a common model to describe phase separation processes of a mixture of two components. We study the Cahn–Hilliard equation coupled with the homogeneous strong anchoring condition (i.e., homogeneous Dirichlet condition) on the relative concentration u of the two phases. Moreover, we adopt no-flux boundary condition to keep conservation of mass. With a specific quartic form of the double-well potential, we prove the existence and uniqueness of the weak solution to this model by interpreting the problem as a gradient flow of the Cahn–Hilliard free energy. Utilizing the minimizing movement scheme and time discretization method, we show that the approximation solutions converge to the weak solution of the Cahn–Hilliard equation. Finally, we prove that the weak solution satisfies an energy dissipation inequality.

Appendix: Ladyzhenskaya’s inequality

The Ladyzhenskaya’s inequality was introduced by O. A. Ladyzhenskaya [41] to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in ${\mathbb {R}}^2$ when the initial data is sufficiently smooth. This inequality is a member of a class of inqualities known as interpolation inequalities.

Lemma 4.3

(Ladyzhenskaya’s inequality) Let $\Omega$ be a Lipschitz domain in ${\mathbb {R}}^d$ ($d=2$ or 3) and $f \in H_0^1(\Omega )$. Then there exists a constant $C > 0$ depending only on $\Omega$ such that

$$\begin{aligned} ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/2} ||\nabla f||_{L^2(\Omega )}^{1/2} \quad \textrm{if}\; d=2, \\ ||f||_{L^4(\Omega )} \le C ||f||_{L^2(\Omega )}^{1/4} ||\nabla f||_{L^2(\Omega )}^{3/4} \quad \textrm{if}\; d=3. \end{aligned}$$