Lₚ-Theory for a Class of Non-Newtonian Fluids

Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier–Stokes problems on domains with compact C³⁻-boundary is proven in an Lₚ-setting for any space dimension n≥2. The stress tensor is assumed to be of the generalized Newtonian type, i.e., =2μ(| E|₂²) E -π I, E=1/2(∇ u+∇ uT), where | E|₂²=∑i,j=1ⁿ εᵢⱼ² denotes the Hilbert–Schmidt norm of the rate of strain tensor E. The viscosity function μ∈ C²⁻( R₊) is subject only to the condition μ(s)>0, μ(s)+2sμ⁽s)>0, s≥ 0, which for the standard power-law–like function μ(s)=μ₀(1+s)d-2/2 merely means μ₀>0 and d≥ 1. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.