##
**Approximation theories for inertial manifolds.**
*(English)*
Zbl 0688.58035

This paper presents a brief review of the existence theories for inertial manifolds via Lyapunov-Perron method, the Hadamard graph transform method and the method of elliptic or parabolic regularization, and three methods for approximating inertial manifolds which can be viewed as modified Galerkin approximation. The advantages and faults of various existence theories of inertial manifolds are discussed from the viewpoint of numerical approximation.

Reviewer: J.Wu

### MSC:

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

35K57 | Reaction-diffusion equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

PDF
BibTeX
XML
Cite

\textit{M. Luskin} and \textit{G. R. Sell}, RAIRO, Modélisation Math. Anal. Numér. 23, No. 3, 445--461 (1989; Zbl 0688.58035)

### References:

[1] | J. E. BILLOTTI, J. P. LASALLE (1971), Dissipative periodic processes, Bull. Amer.Math. Soc, 77, pp. 1082-1088. Zbl0274.34061 MR284682 · Zbl 0274.34061 |

[2] | S.-N. CHOW, K. LU and G. R. SELL (1988), Smoothness of inertial manifolds, Preprint. Zbl0767.58026 MR1180685 · Zbl 0767.58026 |

[3] | P. CONSTANTIN (1988), A construction of inertial manifolds, Preprint. Zbl0691.58040 MR1034492 · Zbl 0691.58040 |

[4] | P. CONSTANTIN, C. FOIAS, B. NICOLAENKO, R. TEMAM (1989), Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, No 70, Springer-Verlag. Zbl0683.58002 MR966192 · Zbl 0683.58002 |

[5] | P. CONSTANTIN, C. FOIAS, B. NICOLAENKO, R. TEMAM (1989), Spectral barriers and inertial manifolds for dissipative partial differential equations, J Dynamics and Differential Equations, 1 (to appear). Zbl0701.35024 MR1010960 · Zbl 0701.35024 |

[6] | P. CONSTANTIN, C. FOIAS, R. TEMAM (1985), Attractors representing turbulent flows, Memoirs Amer. Math. Soc., 314. Zbl0567.35070 MR776345 · Zbl 0567.35070 |

[7] | C. R. DOERING, J. D. GIBBON, D. D. HOLM and B. NICOLAENKO (1988), Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity (to appear). Zbl0655.58021 MR937004 · Zbl 0655.58021 |

[8] | E. FABES, M. LUSKIN and G. R. SELL (1988), Construction of inertial manifolds by elliptic regularization, J. Differential Equations, to appear. Zbl0728.34047 MR1091482 · Zbl 0728.34047 |

[9] | C. FOIAS, M. S. JOLLY, I. G. KEVREKIDIS, G. R. SELL, E. S. TITI (1988), On the computation of inertial manifolds, Physics Letters A, Vol. 131, No 7, 8, pp. 433-436. MR972615 |

[10] | C. FOIAS, B. NICOLAENKO, G. R. SELL, R. TEMAM (1988), Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, pp. 197-226. Zbl0694.35028 MR964170 · Zbl 0694.35028 |

[11] | C. FOIAS, G. R. SELL and R. TEMAM (1986), Inertial manifolds for nonlinear evolutionary equations, IMA Preprint No 234, March, 1986 Also in, J. Differential Equations, 73 (1988), pp. 309-353. Zbl0643.58004 MR943945 · Zbl 0643.58004 |

[12] | C. FOIAS, G. R. SELL and E. S. TITI (1988), Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynamics and Differential Equations, to appear. Zbl0692.35053 MR1010966 · Zbl 0692.35053 |

[13] | C. FOIAS and R. TEMAM (1979), Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, pp. 339-368. Zbl0454.35073 MR544257 · Zbl 0454.35073 |

[14] | J.-M. GHIDAGLIA, Discrétisation en temps et variétés inertielles pour des équations d’évolution aux dérivées partielles non linéaires, Preprint. Zbl0666.35049 · Zbl 0666.35049 |

[15] | J. K. HALE (1988), Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence. Zbl0642.58013 MR941371 · Zbl 0642.58013 |

[16] | J. K. HALE and G. R. SELL (1988), Inertial manifolds for gradient Systems. |

[17] | D. HENRY (1981), Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, No 840, Springer-Verlag, New York. Zbl0456.35001 MR610244 · Zbl 0456.35001 |

[18] | M. S. JOLLY (1988), Explicit construction of an inertial manifold for a reaction diffusion equation, J. Differential Equations (to appear). Zbl0691.35049 MR992147 · Zbl 0691.35049 |

[19] | D. A. KAMAEV (1981), Hopf’s conjecture for a class of chemical kinetics equations, J. Soviet Math., 25, pp. 836-849. Zbl0531.35040 · Zbl 0531.35040 |

[20] | M. LUSKIN and G. R. SELL (1988), Parabolic regularization and the construction of inertial manifolds, Preprint. |

[21] | J. MALLET-PARET (1976), Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22, pp. 331-348. Zbl0354.34072 MR423399 · Zbl 0354.34072 |

[22] | J. MALLET-PARET and G. R. SELL (1987), Inertial manifolds for reaction diffusion equations in higher space dimensions, IMA Preprint No. 331, June 1987, Also in, J. Amer. Math. Soc, 1, No. 4 (1988), pp. 805-866. Zbl0674.35049 MR943276 · Zbl 0674.35049 |

[23] | R. MANÉ (1977), Reduction of semilinear parabolic equations to finite dimensional C1 flows, Geometry and Topology, Lecture Notes in Math., vol. 597, Springer-Verlag, New York, pp.361-378. Zbl0412.35049 MR451309 · Zbl 0412.35049 |

[24] | R. MANÉ, (1981) On the dimension of the compact invariant sets of certain nonlinear maps, Lecture Notes in Math,, vol. 898, Springer-Verlag, New York, pp. 230-242. Zbl0544.58014 MR654892 · Zbl 0544.58014 |

[25] | M. MARION (1988), Inertial manifolds associated to partly dissipative reaction diffusion equations, J. Math. Anal. Appl. (to appear). Zbl0689.58039 · Zbl 0689.58039 |

[26] | X. MORA (1983), Finite dimensional attracting manifolds in reaction diffusion equations, Contemporary Math., 17, pp. 353-360. Zbl0525.35046 MR706109 · Zbl 0525.35046 |

[27] | X. MORA and J. SOLÀ-MORALES (1987), Existence and non-existence of finite dimensional globally attracting invariant manifolds in semilinear damped wave equations, Dynamics of Infinite Dimensional Systems, Springer-Verlag, New York, pp. 187-210. Zbl0642.35062 MR921912 · Zbl 0642.35062 |

[28] | X. MORA and J. SOLÀ-MORALES (1988), The singular limit dynamics of semilinear damped wave equations, Preprint, Univ. Autònoma Barcelona. Zbl0699.35177 MR992148 · Zbl 0699.35177 |

[29] | B. NICOLAENKO, B. SCHEURER and R. TEMAM, (1987), Some global dynamical properties of a class of pattern formation equations, IMA Preprint No. 381. Zbl0691.35019 MR976973 · Zbl 0691.35019 |

[30] | A. PAZY (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol.44, Springer-Verlag, New York. Zbl0516.47023 MR710486 · Zbl 0516.47023 |

[31] | R. J. SACKER (1964), On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, NYU Preprint No. 333, October 1964. |

[32] | R. J. SACKER (1965), A new approach to the perturbation theory of invariant surfaces, Comm. Pure Appl. Math., 18, pp.717-732. Zbl0133.35501 MR188566 · Zbl 0133.35501 |

[33] | R. J. SACKER (1969), A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech., 18, pp.705-762. Zbl0218.34046 MR239221 · Zbl 0218.34046 |

[34] | G. R. SELL and Y. YOU (1988), Inertial manifolds : The non self adjoint case, Preprint. Zbl0760.34051 · Zbl 0760.34051 |

[35] | M. TABOADA (1988), Finite dimensional asymptotic behavior for the Swift-Hohenberg model of convection, Nonlinear Analysis, TMA, to appear. Zbl0707.58019 MR1028246 · Zbl 0707.58019 |

[36] | R. TEMAM (1988), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York. Zbl0662.35001 MR953967 · Zbl 0662.35001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.