EXISTENCE AND UNIQUENESS OF CLASSICAL-SOLUTIONS FOR A CLASS OF COMPLEXITY-2 FLUIDS

In this paper we investigate the existence of classical solutions for a general system of evolution equations and for a corresponding system of steady equations arising in the study of the equations of motion o f incompressible non-Newtonian fluids. Once we have made suitable assu mptions on the differential operators involved, we show existence and uniqueness, for all times, for the abstract evolutionary problem. Usin g the general framework, we carry on our study to a particular class o f complexity-2 fluids, and show that this set of equations admits a un ique, global (in time), classical solution for sufficiently small data . Finally, we establish similar results for the corresponding set of s teady equations.