Existence of weak solutions of the gg-Kelvin–Voight equation

The 2D gg-Navier–Stokes equations have the form ∂u∂t−νΔu+u.∇u+∇p=fin Ω with the continuity equation ∇.(gu)=0in Ω in a bounded domain Ω⊂R2Ω⊂R2 where g=g(x1,x2)g=g(x1,x2) is a smooth real valued function defined on ΩΩ. We use the method described by Roh [J. Roh, gg-Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of gg-Kelvin–Voight equations represented by ∂u∂t−νΔgu+νg(∇g⋅∇)u−αΔgut+αg(∇g⋅∇)ut+u⋅∇u+∇p=f(x) in Ω∇.(gu)=0in Ω We discuss the existence and uniqueness of weak solutions of gg-Kelvin–Voight equations by the use of the well known Feado–Galerkin method.