On the regularity for a class of electro-rheological fluids

We are concerned with a system of nonlinear partial differential equations with p ( x ) -structure, 1 < p ∞ ⩽ p ( x ) ⩽ p 0 < + ∞ , and no-slip boundary conditions. We prove the existence and uniqueness of a C 1 , γ ( Ω ¯ ) ∩ W 2 , 2 ( Ω ) solution corresponding to small data, without further restrictions on the bounds p ∞ , p 0 . In particular this result is applicable to the steady motion of shear-dependent electro-rheological fluids.