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 0.1 $$\begin{aligned} \varepsilon \phi _{tt}+ \phi _t-\Delta (\delta \phi _t-\Delta \phi + g(\phi ))=0, \end{aligned}$$ 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 $$\varepsilon \rightarrow 0^+$$ .