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$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

Radosław Czaja, Piotr Kalita, Alexandre N. Oliveira-Sousa

TL;DR

This work develops a non-autonomous invariant and inertial manifold theory for parabolic-type problems using a fiberwise Lyapunov–Perron approach. By leveraging an exponential splitting of the linear evolution and a generalized gap condition expressed via an admissible norm, it constructs a time-dependent Lipschitz manifold $ig\mathcal{M}(t)ig$ as the graph of a function $ ext{Σ}(t,ullet)$, with controlled growth and an explicit decay rate $oldsymbol{ ext{ω}} \

Abstract

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have $C^1$ regularity.

$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

TL;DR

This work develops a non-autonomous invariant and inertial manifold theory for parabolic-type problems using a fiberwise Lyapunov–Perron approach. By leveraging an exponential splitting of the linear evolution and a generalized gap condition expressed via an admissible norm, it constructs a time-dependent Lipschitz manifold as the graph of a function , with controlled growth and an explicit decay rate $oldsymbol{ ext{ω}} \

Abstract

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have regularity.

Paper Structure

This paper contains 15 sections, 28 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Problem setup and assumptions
  3. Linear evolution process and its exponential splitting.
  4. Renorming of the space .
  5. Nonlinearity, nonlinear process, and the gap condition.
  6. Gronwall lemmas.
  7. Results on existence of Lipschitz invariant and inertial manifolds
  8. Main result.
  9. Fixed point argument.
  10. Definition of the manifold and its invariance.
  11. Cone condition and Lipschitzness of .
  12. Estimate of the distance to the manifold and its controlled growth.
  13. Stable manifold of the invariant Manifold
  14. Differentiable invariant and inertial manifolds
  15. invariant and inertial manifolds

Key Result

Lemma 2.4

If the process $\{L(t,\tau)\colon (t,\tau)\in J\}$ has exponential splitting, then the functions $|\cdot|_{N(\tau)}$ and $|\cdot|_{S(\tau)}$ are norms in $N(\tau)$ and $S(\tau)$, respectively, which are equivalent to $\left\Vert\cdot\right\Vert$ on each of the spaces. Moreover, we have and Furthermore, we have

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Corollary 2.10
  • ...and 59 more