Singular Limits of Voigt Models in Fluid Dynamics

Abstract

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 [33].

Navier–Stokes–Voigt Equations with Memory in 3D Lacking Instantaneous Kinematic Viscosity

“…For every fixed ε > 0 and β > 0, the semigroup S ε,β (t) possesses an exponential attractor E ε,β , and the same is true for the limiting semigroup S 0,0 (t) (see [9]). Again, the task is proving the convergence E ε,β → E 0,0 as ε → 0 and β → 0, in the sense of the symmetric Hausdorff distance.…”

Upper Semicontinuity of Trajectory Attractors for 3D Incompressible Navier–Stokes Equation

“…21) and u(t) is a weak solution of 3D incompressible Navier-Stokes equation, i.e., u(t) ∈ O 0 .From the above argument, we have proved the following proposition.…”

“…There are also a lot of literature on Navier-Stokes-α model [1,2,9,10,18]. Zelati and Gal [21] proved the existence of global and exponential attractors, then they prove the convergence of the (strong) global attractor of the 3D Navier-Stokes-Voight model to the (weak) global attractor of the 3D Navier-Stokes equation, i.e., they prove the convergence of the (strong) global attractor of the 3D Navier-Stokes-Voight model to the trajectory attractor of the 3D Navier-Stokes equation. But, they did not prove upper semicontinuity of trajectory attractors.…”

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

“…Moreover, they showed that as the viscosity coefficient goes to zero, the weak solutions of the Navier-Stokes-Voight equations converge to the weak solutions of the inviscid simplified Bardina model in appropriate norm. The authors in [40] investigated the long-term behavior of the three-dimensional Navier-Stokes-Voigt model as the regularization parameter vanishes and extended the results of [25]. They established the existence of global and exponential attractors of optimal regularity.…”