Regularity results for parabolic systems related to a class of non-Newtonian fluids

We consider a class of parabolic systems of the type: u(t) - div a(x, t, Du) = 0 where the vector field a(x, t, F) exhibits non-standard growth conditions. These systems arise when studying certain classes of non-Newtonian fluids such as electrorheological fluids or fluids with viscosity depending on the temperature. For properly defined weak solutions to such systems, we prove various regularity properties: higher integrability, higher differentiability, partial regularity of the spatial gradient, estimates for the (parabolic) Hausdorff dimension of the singular set. (C) 2003 Elsevier SAS. All rights reserved.