Abstract
We consider heat-conducting viscous incompressible (not necessarily Newtonian) fluids under the general Stokesian constitutive hypotheses. Given a natural and mild condition on the stress tensor at vanishing velocity, which is satisfied for Newtonian fluids, we discuss the stability behavior of stationary states at which the fluid is at rest and at constant temperature. In particular, we prove the existence of global small strong solutions for rather general isothermal non-Newtonian fluids. We also study bifurcation problems and show that subcritical bifurcations can occur. This effect can be seen only if the full energy equation is taken into consideration, that is, if the energy dissipation term is not dropped, as is done in the usual Boussinesq approximation. Bibliography: 29 titles.
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