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Inertial manifolds for partial differential evolution equations under time-discretization: Existence, convergence, and applications. (English) Zbl 0726.35056
Let H be a separable, infinite-dimensional Hilbert space. Denote by A a linear, closed, unbounded and positive operator in H with compact inverse $$A^{-1}$$ and by C a linear, bounded and skew-symmetric operator acting from $$D(A^ s)$$ to H for some $$s\geq 0$$. Suppose that A and C commute. Finally, let F: D(A$${}^{\alpha})\to D(A^{\alpha -\gamma})$$ be a Lipschitz continuous function for some $$\alpha\in R$$, $$0\leq \gamma \leq 1/2.$$
The authors consider evolution equations of the form $(1)\quad u_ t+Au+Cu+F(u)=0\text{ for } t>0$ together with the time-discretized version (fractional step method, $$n=0,1,2,...)$$ $(2)\quad (u^{n+1/2}- u^ n)/h+Au^{n+1}+F(u^ n)=0,\quad (u^{n+1}- u^{n+1/2})/h+C(u^{n+1}+u^{n+1/2})/h=0,$ where h is the time step. The initial condition is $$u(0)=u^ 0\in D(A^{\alpha})$$. It is shown that for sufficiently large d (the spectral gap condition on the eigenvalues of A) a Lipschitz continuous inertial manifold M of dimension d for (1) can be constructed and that, in this case, the discretized version (2) also has an inertial manifold $$M_ h$$ of the same dimension, provided the time step is sufficiently small. Convergence of $$M_ h$$ to M as $$h\to 0$$ in the $$\alpha$$-norm together with an error estimate is established.
The theory is applied to two types of equations: to spatially inhomogeneous complex amplitude equations of the Ginzburg-Landau type and to dissipative perturbations of Korteweg-de Vries equations.

##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 34G20 Nonlinear differential equations in abstract spaces 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35B40 Asymptotic behavior of solutions to PDEs
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