Global existence of solutions for flows of fluids with pressure and shear dependent viscosities (Q1861833)

The equations of a viscous incompressible fluid are considered under the assumption that the viscosity \(\mu\) in the Stokes stress-strain law \(T=\) \(-pI+\) \(\mu D\) depends on the pressure \(p\) and on the invariant \(| D| \equiv\) \(\sqrt{D_{ij}D_{ij}}\) of the rate of strain tensor \(D\) in such a way that \(\partial\mu/\partial p >0\) and \(\partial\mu/\partial | D| <0\). A typical example is given by the formula \(\mu =\) \((\text{const}+\) \(\gamma (p)+\) \(| D| ^2)^{(r-2)/2}\). Here, \(\gamma =\) \(e^{-\alpha p}\) for large values of \(p\), and \(1<r<2\). The existence of weak space periodic solutions is discussed. It is proved that a solution can be obtained globally in time via the limit of a sequence of approximate solutions if they obey some natural a priori estimates.