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 of incompressible non-Newtonian fluids. Once we have made suitable assumptions on the differential operators involved, we show existence and uniqueness, for all times, for the abstract evolutionary problem. Using the general framework, we carry out our study to a particular class of complexity-2 fluids, and show that this set of equations admits a unique, global (in time), classical solution for sufficiently small data. Finally, we establish similar results for the corresponding set of steady equations.