Existence of weak solutions of the -Kelvin–Voight equation

The 2D g -Navier–Stokes equations have the form ∂ u ∂ t − ν Δ u + u . ∇ u + ∇ p = f in  Ω with the continuity equation ∇ . ( g u ) = 0 in  Ω in a bounded domain Ω ⊂ R 2 where g = g ( x 1 , x 2 ) is a smooth real valued function defined on Ω . We use the method described by Roh [J. Roh, g -Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of g -Kelvin–Voight equations represented by ∂ u ∂ t − ν Δ g u + ν g ( ∇ g ⋅ ∇ ) u − α Δ g u t + α g ( ∇ g ⋅ ∇ ) u t + u ⋅ ∇ u + ∇ p = f ( x )  in  Ω ∇ . ( g u ) = 0 in  Ω We discuss the existence and uniqueness of weak solutions of g -Kelvin–Voight equations by the use of the well known Feado–Galerkin method.