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

We discuss solutions u:R3⊃Ω→R3, π:Ω→Rπ: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 μμ. Here u denotes the velocity field, ππ the pressure, σ the stress deviator and f an external volume force. Since we are interested in shear thickening flows μμ 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<1α<1, where Ω0⊂ΩΩ0⊂Ω is an open set with dimH(Ω∖Ω0)≤1dimH(Ω∖Ω0)≤1.