C¹ invariant, stable and inertial manifolds for non-autonomous dynamical systems

2025/07/31 by Radosław Czaja, Czaja, Radosław, Piotr Kalita +3
Engineering · Mathematics · #35B42 #37D10 #37L05 #37L25 #37L30 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2508.00165

openalex publication_date 2025/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C¹ regularity.

Citations

• A Unified Theory for Inertial Manifolds, Saddle Point Property and Exponential Dichotomy
• Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations
• Inertial manifolds and finite-dimensional reduction for dissipative PDEs

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