Unsteady flows of fluids with pressure dependent viscosity in unbounded domains

In order to describe the behavior of various liquid-like materials at high pressures, incompressible fluid models with pressure dependent viscosity seem to be a suitable choice. In the context of implicit constitutive relations involving the Cauchy stress and the velocity gradient these models are consistent with standard procedures of continuum mechanics. Understanding the mathematical properties of the governing equations is connected with various types of idealizations, some of them lead to studies in unbounded domains. In this paper, we first bring up several characteristic features concerning fluids with pressure dependent viscosity. Then we study the three-dimensional flows of a class of fluids with the viscosity depending on the pressure and the shear rate. By means of higher differentiability methods we establish the large data existence of a weak solution for the Cauchy problem. This seems to be a first result that analyzes flows of considered fluids in unbounded domains. Even in the context of purely shear rate dependent fluids of a power-law type the result presented here improves some of the earlier works.