Large data existence theory for unsteady flows of fluids with pressure- and shear-dependent viscosities

A generalization of Navier–Stokes’ model is considered, where the Cauchy stress tensor depends on the pressure as well as on the shear rate in a power-law-like fashion, for values of the power-law index r∈(2dd+2,2]. We develop existence of generalized (weak) solutions for the resultant system of partial differential equations, including also the so far uncovered cases r∈(2dd+2,2d+2d+2] and r=2r=2. By considering a maximal sensible range of the power-law index rr, the obtained theory is in effect identical to the situation of dependence on the shear rate only.