Attractor of the limiting Navier--Stokes--Voigt system in $\mathbb R^4$

TL;DR

This paper develops the theory of global attractors for the Navier--Stokes--Voigt system specifically in four-dimensional space, highlighting unique mathematical properties and providing estimates for attractor dimensions.

Contribution

It establishes well-posedness, dissipativity, and the existence of a global attractor for the Navier--Stokes--Voigt system in four dimensions, a case with unique mathematical simplicity.

Findings

Proved well-posedness of the system in 4D

Established dissipativity and existence of a global attractor

Provided estimates for the dimension of the attractor

Abstract

The Navier--Stokes--Voigt system in the whole four-dimensional space is considered. Although we do not know any physical reasons to consider this system in space dimension four, the attractors theory for this case becomes especially simple and elegant and nothing similar happens when the space dimension is different than four. These notes are devoted to developing this theory, including well-posedness, dissipativity, existence of a global attractor and estimates for its dimension.