ABSTRACT
There are fluids that do not obey the simple relationship between shear stress and shear strain rate given by the following equation for a Newtonian fluid
These fluids have been given the general name non-Newtonian fluids. Many common fluids are non-Newtonian: paints, solutions of various polymers, food products such as apple sauce and ketchup, emulsions of water in oil or oil in water and suspensions of various solids and fibers in a liquid paper pulp or coal slurries and the drilling mud used in well drilling. We shall consider the fluids which exhibit the property to shear thin or shear thicken-the power law fluids. The equations governing the unsteady motion of these fluids have been previously investigated by many authors, e.g. [6]. We are interested in the asymptotic behaviour of the power law fluids. The Cauchy problem of the power-law incompressible fluids in R 3 is described by the system of equations
u=(u 1,…, u 3) represents the velocity field, T is the stress tensor, u 0 is the initial
value of the velocity. The stress tensor is decomposed as
where π is the pressure, δ ij is the Kronecker delta, τ v is the viscous part of the stress. We
will assume the stress tensor τ v of the form
(1.1)
(1.2)
(1.3)
with τ a nonlinear tensor function , where the components of the symmetric deformation velocity tensor are given by
We consider the following growth-conditions
as well as the strong coercivity condition
The existence of a weak solution for and uniqueness and regularity for
n=2, 3 were proved by [6], 1996. There is an extensive literature on decay on the Navier-Stokes equations. We recall
some of them. First result on decay of strong solutions with small initial data was given by Kato [6]. The decay in L 2 norm of weak solution with arbitrary data was proved by Schonbek [14] assuming integrability of the initial data. Further Wiegner [21] and Kajikiya, Miyakawa [5] improved decay replacing assumption on L 1 integrabilty to asumption on L p integrability see [4] resp. that initial data are also initial data of linear heat equation see [15], There are also many results on L 1 decay and decay in Hardy spaces given by Miyakawa see [9]. There are a lot of results of decay in an exterior domain, we can mention e.g. work of Masuda and Hey wood see [4], [7]. Concerning lower bounds we can mention works of Schonbek see [17]–[19]. There is intensive investigation of the boundedness and decay of moments see [20], [2].
