Construction of inertial manifolds by elliptic regularization. (English) Zbl 0728.34047
An inertial manifold for an evolution equation \(u'+Au=F(u)\) on a Hilbert space H is a globally exponentially attracting, positively invariant, finite dimensional Lipschitz manifold \({\mathcal M}\) in H. Sometimes \({\mathcal M}\) can be represented as the graph of a function \(\phi\) : PH\(\to QH\), where P is the orthogonal projection to the space corresponding to the first M eigenvalues \(\lambda_ 1,...,\lambda_ M\) of A and \(Q=I-P\). In that case, \(\Phi\) satisfies \(D\Phi (PF(p+\Phi)-Ap)=QF(p+\Phi)-A\Phi\) on PH. In this paper, the authors solve this first order PDE by considering the elliptic equation \(-\epsilon \Delta \Phi_{\epsilon}+D\Phi_{\epsilon}(PF(p+\Phi_{\epsilon})- Ap)+A\Phi_{\epsilon}=QF(p+\Phi_{\epsilon})\) for \(\epsilon >0\). It is shown that under certain conditions on A and F, \(\Phi_{\epsilon}\) converges to \(\Phi\), provided the ‘spectral gap’ conditions: \(\lambda_{M+1}-\lambda_ M>K_ 1\) and \(\lambda_{M+1}>K_ 2\) are satisfied.
Reviewer: J.So (Edmonton)
MSC:
| 34C30 | Manifolds of solutions of ODE (MSC2000) |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
Keywords:
elliptic regularization; spectral gap condition; inertial manifold; evolution equation; Hilbert spaceReferences:
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