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A simple construction of inertial manifolds under time discretization

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  • 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.

    Mathematics Subject Classification: 35Q10, 65N30.


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