Geodesic
$C^{1,\alpha}$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Tome 259 (1999), pp. 89-121 
P. Kaplitský; J. Málek; J. Stará. $C^{1,\alpha}$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Tome 259 (1999), pp. 89-121. http://geodesic.mathdoc.fr/item/ZNSL_1999_259_a4/
@article{ZNSL_1999_259_a4,
     author = {P. Kaplitsk\'y and J. M\'alek and J. Star\'a},
     title = {$C^{1,\alpha}$-solutions to a class of nonlinear fluids in two dimensions-stationary {Dirichlet} problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {89--121},
     year = {1999},
     volume = {259},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_259_a4/}
}
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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.