Singular limits of Voigt models in fluid dynamics

TL;DR

This paper studies the asymptotic behavior of the 3D Navier-Stokes-Voigt model as the regularization parameter approaches zero, establishing attractors, bounds, and convergence to the classical Navier-Stokes attractor.

Contribution

It provides new results on the existence, bounds, and convergence of attractors for the Navier-Stokes-Voigt model, extending previous work by Kalantarov and Titi.

Findings

Existence of global and exponential attractors with optimal regularity

Explicit upper bounds for attractor dimensions in terms of Grashof number and regularization parameter

Convergence of the Voigt model's attractor to the Navier-Stokes attractor as regularization vanishes

Abstract

We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [31].