Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations
The authors study the viscous Cahn-Hilliard equation, i.e., \[ \varepsilon u_{tt} +u_t+\Delta (\Delta u-u^3+u-\delta u_t)=0,\quad \Delta=\partial^2_x,\;x\in \mathbb{R}.\tag{1} \] The distinguish two cases: (a) \(\varepsilon>0\), \(\delta>0\), (b) \(\varepsilon >0\), \(\delta=0\). The aim is to study (1) in a suitable function space and to prove the existence of an exponential attractor resp. of an inertial manifold. The functional frame for (1) is based on the Sobolev spaces \(H^m=H^m(0,\pi)\cap H_0^1(0, \pi)\), \(H_0^m=H_0^m(0,\pi)\) and \(H^{-m}=(H_0^m)'\); \(\|\;\|_m\) is the norm in \(H^m\). The basic space for (1) is \(X=H^1\times H^{-1}\), i.e., \(u\in H^1\), \(u_t\in H^{-1}\). In Sect. 2, known results are recalled. Thus Theorem 2.1 asserts among others global existence of solutions \((u,u_t)\in C_b([0, \infty);X)\) for initial data \((u_0,u_1 )\in X\), and moreover the existence of an absorbing set \(B=\{(u,v)\in X/ \varphi_0(u,v)\leq R\}\), some \(R>0\) where \(\varphi_0\) is a certain functional of somewhat involved structure which is bounded from below. Theorem 2.2 then asserts the existence of global attractors \(A_{\varepsilon \delta}\) and \(A_{\varepsilon 0}\) for the two cases (a), (b). In Sect. (3) the notion of exponential attractor is explained and, based on \textit{A. Eden}, \textit{C. Foias}, \textit{B. Nicolaenko} and \textit{R. Temam} [Exponential attractors for dissipative evolution equations. (Chichester: Wiley) (1994; Zbl 0842.58056)] in Theorems 2.1, 2.2, the existence of an exponential attractor for (1) is proved. The difficulty is to show that the attractor is compact. This is achieved by restricting the flow \(S(t)\) for (1) to more regular data, inducing more regular trajectories and thus allowing for compact embeddings. Moreover the so-called discrete squeezing property has to be checked. The second half of the paper is devoted to the proof that (1) admits an inertial manifold. The proof splits into two steps. First it is assumed that the nonlinearity \(F(U)\) satisfies a global Lipschitz condition. This assumption, not satisfied by (1), is removed in a second step via a standard smoothing procedure. After a rescaling, (1) is cast into an evolution equation \(U_t+AU =F(U)\), \(U \in X\) where the linear operator \(A\) is defined in terms of the linear part of (1). The major difficulty to overcome to show that \(A\) satisfies the well known gap condition, which in turn implies the existence of an inertial manifold. This requires very subtle and difficult spectral estimates. The paper, excellently written, can be understood by someone not specifically familiar with the subject.
scientific article; zbMATH DE number 2104892
scientific article