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October  1997, 3(4): 531-540. doi: 10.3934/dcds.1997.3.531

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, China

Received  January 1996 Published  July 1997

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.

Keywords: Nonlinear evolution equation, time discretization., inertial manifold.
Mathematics Subject Classification: 35Q10, 65N3.
Citation: Changbing Hu, Kaitai Li. A simple construction of inertial manifolds under time discretization. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 531-540. doi: 10.3934/dcds.1997.3.531
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