Multivalued Attractors and their Approximation: Applications to the Navier-Stokes equations

TL;DR

This paper investigates multivalued attractors and their approximation methods, applying the theory to the Navier-Stokes equations, demonstrating convergence of discrete attractors to the continuous system's attractor as discretization improves.

Contribution

It develops an abstract framework for multivalued semigroups and applies it to prove convergence of numerical attractors for the Navier-Stokes equations.

Findings

Discrete attractors converge to continuous attractors as time-step decreases.

The implicit Euler scheme effectively approximates the Navier-Stokes attractor.

The framework can be applied to other set-valued dynamical systems.

Abstract

This article is devoted to the study of multivalued semigroups and their asymptotic behavior, with particular attention to iterations of set-valued mappings. After developing a general abstract framework, we present an application to a time discretization of the two-dimensional Navier-Stokes equations. More precisely, we prove that the fully implicit Euler scheme generates a family of discrete multivalued dynamical systems, whose global attractors converge to the global attractor of the continuous system as the time-step parameter approaches zero.