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.

Keywords

• navier–stokes–voigt equations
• viscoelastic models
• non-newtonian fluid
• strong solutions
• existence and uniqueness theorem
• faedo–galerkin approximations
• stokes operator
• long-time behavior