Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space

The present article is intended to study the boundedness of solutions for an unsteady non-Newtonian flow, whose strain-stress relationship is provided by the Sisko fluid model. Such kind of flow can appear in different scenarios, but we make it particular to the field of magnetohydrodynamics, as it is a remarkable area of research in its own right. The ideas exposed in this work can be extended, in a similar manner, to a wide range of applications. To make our fluid further general, we use the Darcy's law to characterize the porous space in which the fluid is flowing. We develop the boundedness criteria provided that the velocity w and the function g = -dw/dr satisfy w, g, dg/dr, d2g/dr2 is an element of L2(0, T; BMO), where BMO means bounded mean oscillation space. For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.