Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions

We study a planar flow of a generalized Newtonian fluid under the Dirichlet boundary condition. The fluid is characterized by a nonlinear dependence of the stress tensor on the symmetric part of the velocity gradient. We prove that the unique weak solution of this problem has a Holder continuous gradient provided the growth of the stress tensor is of order p - 1 for a certain p E (2, 4). The result is global in time and in space.