On the Kneser property of the three-dimensional Navier-Stokes equations with damping

TL;DR

This paper investigates the topological properties of the set of solutions to the 3D Navier-Stokes equations with damping, focusing on connectedness, compactness, and the structure of the global attractor.

Contribution

It provides new results on the connectedness and regularity of the global attractor, and offers a novel proof of its existence for strong solutions.

Findings

Connectedness of the global attractor for certain damping parameters

Compactness and regularity results for the attainability set

A new proof of the existence of the global attractor for strong solutions

Abstract

In this paper, we study the connectedness and compactness of the attainability set of weak solutions to the three-dimensional Navier--Stokes equations with damping. Depending on the value of the parameter \b{eta}, which controls the damping term, we establish these results with respect to either the weak or the strong topology of the phase space. In the latter case, we also prove that the global attractor is connected. Additionally, we establish results concerning the regularity of the global attractor and provide a new proof of its existence for strong solutions.