Inertial manifolds and normal hyperbolicity
This is a rather long technical digression on inertial manifolds of evolution equations of the form: \[ u_t+Au= f(u). \] Here \(-A\) generates a strongly continuous semigroup \(V_t\), \(t\geq 0\) on an underlying Banach space \(E_0\). There are two further Banach spaces \(E\), \(F\) related to \(E_0\) by the continuous embeddings \(E\subset F\subset E_0\). Various hypothesis are imposed on \(f\), \(A\) so as to secure the existence of inertial manifolds. In a first setting, \(f\) is globally Lipschitz from \(E\) to \(F:|f(u)-f(v)|_F\leq M_1|u-v|_E\). Furthermore, one is given numbers \(0<\lambda_n\leq\Lambda_n\), \(n\geq 1\), finite-dimensional projection operators \(P_n\), \(n\geq 1\), a number \(\alpha\in(0,1)\) and constants \(K_1\), \(K_2\) such that the following holds (where \(Q_n=1-P_n\)): (1) \(P_nE_0\) is invariant under \(V_t\), \(t\geq 0\), and \(V_tP_n\), \(t\geq 0\) extends into a group \(V_tP_n\), \(t\in\mathbb{R}\) on \(P_nE_0\) such that \(|V_tP_n|_{L(E)}\leq K_1e^{-\lambda_nt}\), \(t\leq 0\) and \(|V_tP_n|_{L(F,E)}\leq K_2\lambda^\alpha_ne^{-\lambda_nt}\), \(t\leq 0\), (2) \(Q_nE\) is invariant under \(V_t\), \(t\geq 0\) with \(|V_tQ_n|_{L(E)}\leq K_2e^{-\Lambda_nt}\), \(t\geq 0\), \(|V_tQ_n|_{L(E,F)}\leq K_2(t^{-\alpha}+\Lambda^\alpha_n)e^{-\Lambda_nt}\), \(t\geq 0\). The assumption \(V_tF\subset E\), \(t>0\) is incorporated in (1), (2). Based on this framework, the existence of an inertial manifold can be proved under the assumption that a spectral gap condition is satisfied. A number of properties of the inertial manifolds are proved, among others asymptotic completeness. At a later stage, the assumption that \(f\) is globally Lipschitz is removed by a well-known cut off procedure. A series of further properties such as smoothness are discussed. For reasons of space we cannot do justice to this rather long paper. However, we have to point out that the paper can be highly recommended to everybody interested in inertial manifolds and the techniques of their construction. Moreover, there are a series of technical improvements in the paper as far as the assumptions on the nonlinearity \(f\) are concerned, which are of interest to specialists in the field. The paper, although technical, is written in a readable way, proofs are carried out in detail and the whole topic is presented in an essentially selfcontained way.
scientific article; zbMATH DE number 943757
scientific article