A simple construction of inertial manifolds under time discretization

Changbing Hu; Kaitai Li

doi:10.3934/dcds.1997.3.531

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1997, Volume 3, Issue 4: 531-540. Doi: 10.3934/dcds.1997.3.531

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A simple construction of inertial manifolds under time discretization

Changbing Hu^1, and
Kaitai Li^1,

1.
Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049

Received: December 31, 1995

Published: September 1997

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Abstract

In this article, we obtain the existence of inertial manifolds under time discretization based on their invariant property. In [1], the authors gave their existence by finding the fixed point of some inertial mapping defined by a sum of infinite series:

$ T_h^0\Phi(p)=\sum_{k=1}^{\infty}R(h)^kQF(p^{-k}+\Phi(p^{-k})) $

where $p^{-k}=(S^h_\Phi)^{-k}(p)$, see [1] for detailed definition. Here we get the existence by solving the following equation about $\Phi$:

$\Phi(S_\Phi^h(p))=R(h)[\Phi(p)+hQF(p+\Phi(p))] \mbox{ for }\forall p\in PH.$

See section 1 for further explanation which describes just the invariant property of inertial manifolds. Finally we prove the $C^1$ smoothness of inertial manifolds.

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Mathematics Subject Classification: 35Q10, 65N30.

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