Analysis of generalized Navier–Stokes equations for stationary shear thickening flows

Abstract

We discuss solutions u:R3⊃Ω→R3, π:Ω→R to generalized Navier–Stokes equations divσ=(∇u)u+∇π−f,σ=σ(ε(u))=μ(|ε(u)|)ε(u), with generalized viscosity function μ. Here u denotes the velocity field, π the pressure, σ the stress deviator and f an external volume force. Since we are interested in shear thickening flows μ is assumed to be increasing but we do not assume any growth condition. The result is the existence of a weak solution to the equation above with V(ε(u))∈Wloc1,2(Ω), where V(ε)=∫0|ε|μ(s)ds. Moreover, we have u∈C1,α(Ω0,R3) for any α<1, where Ω0⊂Ω is an open set with dimH(Ω∖Ω0)≤1.