Frequency theorem for parabolic equations and its relation to inertial manifolds theory
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911772238151680 |
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| author | Anikushin, Mikhail |
| author_facet | Anikushin, Mikhail |
| contents | We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R. A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_12031 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Frequency theorem for parabolic equations and its relation to inertial manifolds theory Anikushin, Mikhail Analysis of PDEs Dynamical Systems Optimization and Control 35K58, 35B42, 37L45, 37L15 We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R. A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field. |
| title | Frequency theorem for parabolic equations and its relation to inertial manifolds theory |
| topic | Analysis of PDEs Dynamical Systems Optimization and Control 35K58, 35B42, 37L45, 37L15 |
| url | https://arxiv.org/abs/2011.12031 |