$C^{1,\alpha}$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem

We prove the global existence of $C^{1,\alpha}$-solutions to a system of nonlinear equations describing steady planar motions of a certain class of non-Newtonian fluids including in particular various variants of the power-law models. We study the Dirichlet problem. The nonlinear operator has a $p$-potential structure. If $p>3/2$ we construct global $C^{1,\alpha}$-solutions up to the boundary, while for $p>6/5$ solutions with interior $C^{1,\alpha}$-regularity are obtained. A proof of global higher regularity is outlined. Uniqueness of $C^{1,\alpha}$-solutions within the class of weak solutions is also proved assuming the smallness of data.