Global existence of solutions for flows of fluids with pressure and shear dependent viscosities

There is clear and incontrovertible evidence that the viscosity of many liquids depends on the pressure. While the density, as the pressure is increased by orders of magnitude, suffers small changes in its value, the viscosity changes dramatically. It can increase exponentially with pressure. In many fluids, there is also considerable evidence for the viscosity to depend on the rate of deformation through the symmetric part of the velocity gradient, and most fluids shear thin, i.e., viscosity decreases with an increase in the rate of shear. In this paper, we study the flow of fluids whose viscosity depends on both the pressure and the symmetric part of the velocity gradient. We find that the shear thinning nature of the fluid can be gainfully exploited to obtain global existence of solution, which would not be possible otherwise. Previous studies of fluids with pressure dependent viscosity require strong restrictions to all data, or assume forms that are clearly contrary to experiments, namely that the viscosity decreases with the pressure. We are able to establish existence of space periodic solutions that are global in time for both the two- and three-dimensional problem, without restricting ourselves to small data.