### Abstract

This paper discusses the existence and the convergence of inertial manifolds for approximations to semi-linear evolution equations in Banach spaces. Our approximation considered here is closely related to Chernoff’s product formulas. It is shown that the approximation possesses an inertial manifold and this manifold converges to the inertial manifold for the evolution equation. A "parabolic" version of Chernoff’s lemma is established and used to prove the convergence theorems. As an application the schemes of "Crank-Nicholson type" are considered. Finally, the existence of inertial manifolds for the evolution equation is discussed under the condition that the approximations possess inertial manifolds.

Original language | English |
---|---|

Pages (from-to) | 1117-1134 |

Number of pages | 18 |

Journal | Differential and Integral Equations |

Volume | 8 |

Issue number | 5 |

Publication status | Published - 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Differential and Integral Equations*,

*8*(5), 1117-1134.

**Convergence and approximation of inertial manifolds for evolution equations.** / Kobayashi, Kazuo; Aftabizadeh, Reza.

Research output: Contribution to journal › Article

*Differential and Integral Equations*, vol. 8, no. 5, pp. 1117-1134.

}

TY - JOUR

T1 - Convergence and approximation of inertial manifolds for evolution equations

AU - Kobayashi, Kazuo

AU - Aftabizadeh, Reza

PY - 1995

Y1 - 1995

N2 - This paper discusses the existence and the convergence of inertial manifolds for approximations to semi-linear evolution equations in Banach spaces. Our approximation considered here is closely related to Chernoff’s product formulas. It is shown that the approximation possesses an inertial manifold and this manifold converges to the inertial manifold for the evolution equation. A "parabolic" version of Chernoff’s lemma is established and used to prove the convergence theorems. As an application the schemes of "Crank-Nicholson type" are considered. Finally, the existence of inertial manifolds for the evolution equation is discussed under the condition that the approximations possess inertial manifolds.

AB - This paper discusses the existence and the convergence of inertial manifolds for approximations to semi-linear evolution equations in Banach spaces. Our approximation considered here is closely related to Chernoff’s product formulas. It is shown that the approximation possesses an inertial manifold and this manifold converges to the inertial manifold for the evolution equation. A "parabolic" version of Chernoff’s lemma is established and used to prove the convergence theorems. As an application the schemes of "Crank-Nicholson type" are considered. Finally, the existence of inertial manifolds for the evolution equation is discussed under the condition that the approximations possess inertial manifolds.

UR - http://www.scopus.com/inward/record.url?scp=84972513429&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972513429&partnerID=8YFLogxK

M3 - Article

VL - 8

SP - 1117

EP - 1134

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 5

ER -