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.


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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