Singular limits of Voigt models in fluid dynamics

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].

v2: the statement and proof of Theorem 5.2 have changed. In the previous version, the proof was based on Theorem 4.9 in a book of Ladyzhenskaya, which turns out to be based on a faulty assumption. See Remark 5.5 for details

References in corpus (3)

• Analysis of a General Family of Regularized Navier-Stokes and MHD Models
• Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations
• Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

Cited by in corpus (2)

• Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity
• Uniformly attracting limit sets for the critically dissipative SQG equation