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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.

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
Full Text: DOI
[1] Beyn, W.J, On the numerical approximation of phase portraits near stationary points, SIAM J. numer. anal., 24, 1095-1113, (1987) · Zbl 0632.65083
[2] Constantin, P; Foias, C; Nicolaenko, B; Temam, R, Nouveaux résultats sur LES variétés inertielles pour LES équations différentielles dissipatives, C. R. acad. sci. Paris Sér. I math., 302, 375-378, (1986) · Zbl 0591.35064
[3] Constantin, P; Foias, C; Temam, R; Constantin, P; Manley, O, Determining modes and fractal dimension of turbulent flows, Mem. amer. math. soc., J. fluid. mech., 150, No. 314, 427-440, (1985) · Zbl 0607.76054
[4] Conway, E; Hoff, D; Smoller, J, Large time behavior of solutions of nonlinear reaction-diffusion equations, SIAM J. appl. math., 35, 1-16, (1978) · Zbl 0383.35035
[5] {\scF. Demengel and J. M. Ghidaglia}, Construction of inertial manifolds via the Lyapunov-Perron method, to appear. · Zbl 0723.58033
[6] Doering, C.R; Elgin, J.N; Gibbon, J.D; Holm, D.D, Finite dimensionality in the laser equations in the good cavity limit, Phys. lett. A, 129, 310-316, (1988)
[7] {\scC. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko}, Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, to appear. · Zbl 0655.58021
[8] Foias, C; Manley, O; Temam, R; Treve, Y, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 6, (1983) · Zbl 0584.35007
[9] {\scC. Foias, B. Nicolaenko, G. R. Sell, and R. Temam}, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., to appear · Zbl 0591.35063
[10] Foias, C; Prodi, G, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, (), 1-34 · Zbl 0176.54103
[11] Foias, C; Sell, G.R; Temam, R; Foias, C; Sell, G.R; Temam, R, Inertial manifolds for nonlinear evolutionary equations, J. differential equations, C. R. acad. sci. Paris I math., 301, 139-141, (1985) · Zbl 0591.35062
[12] Foias, C; Temam, R, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. math. pures appl., 58, 339-368, (1979) · Zbl 0454.35073
[13] {\scJ. M. Ghidaglia and B. Héron}, Dimension of the attractors associated to the Ginzburg-Landau partial differential equations, Physica D{\bf28} 282-304. · Zbl 0623.58049
[14] Hale, J.K; Lin, X.B; Raugel, G, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. comp., 50, 89-123, (1988) · Zbl 0666.35013
[15] Mallet-Paret, J; Sell, G.R, Inertial manifolds for reaction-diffusion equations in higher space dimensions, (), Minneapolis · Zbl 0674.35049
[16] Nicolaenko, B; Scheurer, B; Temam, R, Some global dynamical properties of a class of pattern formation equations, (), Minneapolis · Zbl 0691.35019
[17] Nirenberg, L, On elliptic partial differential equations, Ann. scuola norm. sup. Pisa, 13, 116-162, (1959) · Zbl 0088.07601
[18] Richards, J, On the gaps between numbers which are the sum of two squares, Adv. in math., 46, 1-2, (1982) · Zbl 0501.10047
[19] Shub, M, Global stability of dynamical systems, (1987), Springer-Verlag New York
[20] Scheurer, B, Quelques propriétés dynamiques globales des équations de Kuramoto-Sivashinsky et de Cahn-Hilliard, () · Zbl 0654.58022
[21] Temam, R, Infinite dimensional dynamical system in mechanics and physics, (1988), Springer-Verlag New York
[22] Demengel, F; Ghidaglia, J.M, Discrétisation en temps et variétés inertielles pour des équations d’évolution aux dérivés partielles non linéaires, C. R. acad. sci. Paris Sér. I, 307, 453-458, (1988) · Zbl 0666.35049
[23] Constantin, P; Foias, C; Nicolaenko, B; Temam, R, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. dynamics and differential equations, 1, 45-73, (1989) · Zbl 0701.35024
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