YOUNG MEASURE-VALUED SOLUTIONS FOR NON-NEWTONIAN INCOMPRESSIBLE FLUIDS

For the model of a nonlinear bipolar fluid, in which the highest order viscosity vanishes, and the viscous part of the stress tenser satisfies a growth condition of the form \tau(ij)(e)\ less than or equal to C(1+\e\)(p-1), C > 0, e the rate of strain tenser, Ne demonstrate the existence of Young-measure valued solutions for p > 1 (in dim n = 2) and for p > 6/5 (in dim n = 3); these solutions are proven to be weak solutions for 3/2 < p < 2 (in dim n = 2) and for 9/5 < p < 11/5 (in dim n = 3) and unique regular weak solutions for p greater than or equal to 2 (in dim n = 2) and for p greater than or equal to 11/5 (in dim n = 3). Much of the analysis deals with the associated space periodic problems.