Some remarks on inertial manifolds.

*(English)*Zbl 0815.35037The aim of the author is to construct inertial manifolds for a class of evolution equations of the form (1) \(u_ t + Au + R(t,u) = 0\) which takes place on a Hilbert space \(H\) with norm \(\|\;\|\). In order to assure the existence of such manifolds the author imposes some assumptions on (1) which are to some extent weaker than those usually encountered. \(A\) is assumed to be a positive selfadjoint operator with compact resolvents and spectrum \(0 < \lambda_ 1 \leq \lambda_ 2 \dots\); we denote by \(w_ j\) an eigenvector of \(\lambda_ j\). Crucial to the theory is a spectral gap condition of the form (2) \(\lambda_{N+1} - \lambda_ N > K_ 1(\lambda^{\gamma/2}_{N+1} + \lambda_ N^{\gamma/2})^ 2\) for some \(N > 0\). The constant \(K_ 1\) and \(\gamma \in (0,{1 \over 2})\) are determined by the nonlinearity \(R(t,u)\) which is assumed to satisfy the following conditions:

(3) \(\| R(t,u) \| \leq K_ 0\), (4) \(\| R(t,u) - R(t,v) \| \leq K_ 1 \| A^ \gamma (u - v) \|\), (5) \(\| R(t + h,u) - R(t,u) \| \leq K_ 2 | h |\),

where \(u,v \in \text{dom} (A^ \gamma)\). Under these assumptions the evolution equation (1) generates a semiflow \(S(t,t_ 0)\) which maps \(\text{dom} (A^ \gamma)\) into itself and which has some smoothness properties which are proved in the course of the paper. Let \(P_ N\) be the orthogonal projection onto \(\text{span} (w_ 1, \dots, w_ N)\). The first main result (theorem 1) states that there is a Lipschitz mapping \(\varphi (\cdot, \cdot)\) from \(\mathbb{R} \times P_ N \text{dom} (A^ \gamma)\) into \((1-P_ N) \text{dom} (A^ \gamma)\) such that for each \(t \in\mathbb{R}\) the manifold \(M_ t = \text{graph} (\varphi (t, \cdot))\) has the following properties: (a) \(M_ t = S(t,t_ 0) M_{t_ 0}\), \(t,t_ 0 \in\mathbb{R}\), (b) if \(u(t) = S(t,t_ 0) u_ 0\) then there exists \(v_ 0 \in M_{t_ 0}\) such that \[ \Bigl \| A^ \gamma \bigl( S(t + t_ 0, t_ 0) u_ 0 - S(t + t_ 0, t_ 0) v_ 0 \bigr) \Bigr \| \leq C_ 1e^{-t\nu},\;t \geq 0 \] for suitable \(C_ 1\), \(\nu > 0\) independent of \(t_ 0,t\). There are further results of perturbational nature which compare the manifolds \(M_ t\) with \(\widetilde M_ t\), \(t \in\mathbb{R}\) which emerge from nonlinearities \(R(t,u)\) and \(\widetilde R(t,u)\) respectively, and which loosely speaking express that \(M_ t\) and \(\widetilde M_ t\) are asymptotically close if \(R(t,u)\) and \(\widetilde R(t,u)\) are asymptotically close. The proofs are based on a series of technical lemmas. Among these, lemma 3.3 in particular expresses that the flow \(S(t,t_ 0)\) has a certain squeezing property which differs from that usually encountered.

(3) \(\| R(t,u) \| \leq K_ 0\), (4) \(\| R(t,u) - R(t,v) \| \leq K_ 1 \| A^ \gamma (u - v) \|\), (5) \(\| R(t + h,u) - R(t,u) \| \leq K_ 2 | h |\),

where \(u,v \in \text{dom} (A^ \gamma)\). Under these assumptions the evolution equation (1) generates a semiflow \(S(t,t_ 0)\) which maps \(\text{dom} (A^ \gamma)\) into itself and which has some smoothness properties which are proved in the course of the paper. Let \(P_ N\) be the orthogonal projection onto \(\text{span} (w_ 1, \dots, w_ N)\). The first main result (theorem 1) states that there is a Lipschitz mapping \(\varphi (\cdot, \cdot)\) from \(\mathbb{R} \times P_ N \text{dom} (A^ \gamma)\) into \((1-P_ N) \text{dom} (A^ \gamma)\) such that for each \(t \in\mathbb{R}\) the manifold \(M_ t = \text{graph} (\varphi (t, \cdot))\) has the following properties: (a) \(M_ t = S(t,t_ 0) M_{t_ 0}\), \(t,t_ 0 \in\mathbb{R}\), (b) if \(u(t) = S(t,t_ 0) u_ 0\) then there exists \(v_ 0 \in M_{t_ 0}\) such that \[ \Bigl \| A^ \gamma \bigl( S(t + t_ 0, t_ 0) u_ 0 - S(t + t_ 0, t_ 0) v_ 0 \bigr) \Bigr \| \leq C_ 1e^{-t\nu},\;t \geq 0 \] for suitable \(C_ 1\), \(\nu > 0\) independent of \(t_ 0,t\). There are further results of perturbational nature which compare the manifolds \(M_ t\) with \(\widetilde M_ t\), \(t \in\mathbb{R}\) which emerge from nonlinearities \(R(t,u)\) and \(\widetilde R(t,u)\) respectively, and which loosely speaking express that \(M_ t\) and \(\widetilde M_ t\) are asymptotically close if \(R(t,u)\) and \(\widetilde R(t,u)\) are asymptotically close. The proofs are based on a series of technical lemmas. Among these, lemma 3.3 in particular expresses that the flow \(S(t,t_ 0)\) has a certain squeezing property which differs from that usually encountered.

Reviewer: B.Scarpellini (Basel)