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Frequency theorem for parabolic equations and its relation to inertial manifolds theory

Mikhail Anikushin

Abstract

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.

Frequency theorem for parabolic equations and its relation to inertial manifolds theory

Abstract

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.

Paper Structure

This paper contains 11 sections, 13 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Optimization of quadratic functionals
  3. Relaxing the controllability
  4. Realification of the operator $P$ and its sign properties
  5. Inertial manifolds for semilinear parabolic equations and frequency-domain methods
  6. Preliminaries
  7. Optimality of the frequency inequality
  8. Spectral Gap Condition
  9. Other frequency-domain conditions
  10. The Circle Criterion
  11. An example

Key Result

Theorem 1

Let $A$ be also the generator of a $C_{0}$-semigroup $\mathbb{H}_{\alpha}$ and let (RES) be satisfied. Then the following conditions for the pair $(A,B)$ and the form $\mathcal{F}$ are equivalent: 1. For some $\delta'>0$ we have $\mathcal{F}(-(A - i\omega I)B\xi,\xi) \leq -\delta' |\xi|^{2}_{\Xi}$ f where $v(t)=v(t,v_{0},\xi)$ is the solution to EQ: ControlSystem with arbitrary $v(0)=v_{0} \in \ma

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 2
  • proof
  • ...and 16 more