Uniqueness and regularity of flows of non-Newtonian fluids with critical power-law growth

Abstract

We deal with flows of non-Newtonian fluids in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index [Formula: see text], we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices [Formula: see text]. Such a result was available for [Formula: see text] and therefore the paper fills the gap and extends the uniqueness result to the whole range of [Formula: see text]’s for which the energy equality holds.