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Dynamics of the viscous Cahn-Hilliard equation. (English) Zbl 1151.35008
The viscous Cahn-Hilliard problem of the form: \[ \begin{gathered} (1-\nu) u_t= -\Delta(\Delta u+ f(u)- \nu u_t)\quad\text{in }\Omega,\\ u(t,x)=\Delta u(t, x)= 0\quad\text{in }\partial\Omega,\\ u(0,x)= u_0(x)\end{gathered}\tag{1.1} \] is considered. There is assumed, that \(\nu\in[0, 1]\), \(f\in C^1(\mathbb{R},\mathbb{R})\) satisfies suitable growth and dissipation conditions and \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\), \(n\geq 3\).
The aim of the paper is to study the nonlinear semigroup generated by (1.1) on the phase space \(H^1_0(\Omega)\) (and on \(W^{1,p}_0(\Omega)\) in case of local existence and regularity of solutions) with the equation satisfied in \(H^{-1}(\Omega)\) when \(\nu> 0\) or in \(H^{-3}(\Omega)\) when \(\nu= 0\). The main result states that the asymptotic dynamics of (1.1) is the same in a suitable sense for all values of \(\nu\) in \([0,1]\); the global attractors \({\mathcal A}_\nu\) are shown to be continuous for \(\nu\in[0, 1]\). The asymptotic dynamics of the semilinear heat equation \((\nu= 1)\) is thus the same as the asymptotic dynamics of the Cahn-Hillard equation \((\nu= 0)\). Also the Hausdorff dimension of the global attractors \({\mathcal A}_\nu\) for (1.1) is shown to be independent from \(\nu\).

35B41 Attractors
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI
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