Existence and Continuity of Inertial Manifolds for the Hyperbolic Relaxation of the Viscous Cahn–Hilliard Equation

Abstract

We consider the hyperbolic relaxation of the viscous Cahn–Hilliard equation

$$\begin{aligned} \varepsilon \phi _{tt}+ \phi _t-\Delta (\delta \phi _t-\Delta \phi + g(\phi ))=0, \end{aligned}$$

(0.1)

in a bounded domain of ${\mathbb R}^d$ with smooth boundary when subject to Neumann boundary conditions, and on rectangular domains when subject to periodic boundary conditions. The space dimension is $d=1,$ 2 or 3, but it is required $\delta =\varepsilon =0$ when $d=2$ or 3; $\delta$ being the viscosity parameter. The constant $\varepsilon \in (0,1]$ is a relaxation parameter, $\phi$ is the order parameter and $g:{\mathbb R}\rightarrow {\mathbb R}$ is a nonlinear function. This equation models the early stages of spinodal decomposition in certain glasses. Assuming that $\varepsilon$ is dominated from above by $\delta$ when $d=2$ or 3, we construct a family of exponential attractors for Eq. (0.1) which converges as $(\varepsilon ,\delta )$ goes to $(0,\delta _0),$ for any $\delta _0\in [0,1],$ with respect to a metric that depends only on $\varepsilon$, improving previous results where this metric also depends on $\delta$. Then we introduce two change of variables and corresponding problems, in the case of rectangular domains and $d=1$ or 2 only. First, we set $\tilde{\phi }(t)=\phi (\sqrt{\varepsilon } t)$ and we rewrite Eq. (0.1) in the variables $(\tilde{\phi },\tilde{\phi }_t).$ We show that there exist an integer n, independent of both $\varepsilon$ and $\delta$, a value $0<\tilde{\varepsilon }_0(n)\le 1$ and an inertial manifold of dimension n, for either $\varepsilon \in (0,\tilde{\varepsilon }_0]$ and $\delta =2\sqrt{\varepsilon }$ or $\varepsilon \in (0,\tilde{\varepsilon }_0]$ and $\delta \in [0,3\varepsilon ]$. Then, we prove the existence of an inertial manifold of dimension that depends on $\varepsilon$, but is independent of $\delta$ and $\eta$, for any fixed $\varepsilon \in (0,(\eta -2)^2]$ and every $\delta \in [\varepsilon ,(2-\eta )\sqrt{\varepsilon }]$, for an arbitrary $\eta \in (1,2)$. Next, we show the existence of an inertial manifold of dimension that depends on $\varepsilon$ and $\eta$, but is independent of $\delta$, for any fixed $\varepsilon \in (0,\frac{1}{(2+\eta )^2}]$ and every $\delta \in [(2+\eta )\sqrt{\varepsilon },1]$, where $\eta >0$ is arbitrary chosen. Moreover, we show the continuity of the inertial manifolds at $\delta =\delta _0,$ for any $\delta _0\in [0,(2-\eta )\sqrt{\varepsilon }]\cup [(2+\eta )\sqrt{\varepsilon },1]$. Second, we set $\phi _t=-(2\varepsilon )^{-1}(I-\delta \Delta )\phi +\varepsilon ^{-1/2}v$ and we rewrite Eq. (0.1) in the variables $(\phi ,v)$. Then, we prove the existence of an inertial manifold of dimension that depends on $\delta$, but is independent of $\varepsilon ,$ for any fixed $\delta \in (0,1]$ and every $\varepsilon \in (0,\frac{3}{16}\delta ^2]$. In addition, we prove the convergence of the inertial manifolds when $\varepsilon \rightarrow 0^+$.