Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Abstract

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

Numerical analysis of a corrected Smagorinsky model

“…The existence and uniqueness of strong solutions of the incompressible Navier-Stokes-Voigt model is studied in [3].…”

“…The new term in our model has similarity to the Voigt term used in Voigt/Kelvin‐Voigt/Kelvin Model [41] for viscoelastic fluids. There has been lot of recent works on Voigt Model, see for example [3, 21, 24, 25]. Recently, Rong et al [38] and Berselli et al [4] all studied the extension of the Baldwin and Lomax model [2] to non‐equilibrium ($$$ \frac{d}{dt}\overline{{\left\Vert {u}^{\prime}\right\Vert}^2}\ne 0 $$$, for a precise definition see equilibrium) problems.…”

Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

“…Using techniques similar to the ones in the proof of Theorem 1 from [8], it can be shown that problem (26) has a unique solution (v, p) in the space E(Q T ). Consequently, equation (24) is solvable.…”

“…• The Navier-Stokes-Voigt equations (Kelvin-Voigt type viscoelastic fluids) when δ = 1, µ 0 > 0, µ 1 > 0, η ≡ 0 (see [6][7][8][9]); •…”

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

“…In the literature, some authors have referred the system ()–() as system of Navier–Stokes–Voigt equations (see, e.g., Baranovskii 25 and Kalantarov and Titi 26 ), since the system can be considered as a regularization of Navier–Stokes equations by ϰ 27 . Various inverse problems for the Navier–Stokes equations were considered in many works, for example, previous works 6,7,28–33 and references therein.…”