EXISTENCE AND REGULARITY OF SOLUTIONS AND THE STABILITY OF THE REST STATE FOR FLUIDS WITH SHEAR DEPENDENT VISCOSITY

Abstract

In this paper we clarify and discuss some subtle features concerning the non-Newtonian fluid models which were considered by Málek, Nečas and Růžička.19 We establish new results regarding the stability of the rest state of mechanically isolated flows of such non-Newtonian fluids for arbitrary initial disturbances. We also discuss some results concerning the existence and regularity of solutions for small data. These results are based on a new method, giving convergence almost everywhere of approximations of gradients from boundedness of fraction of the L2-norm of the second and first derivatives, developed by Nečas to study existence of solutions (global in time) to a class of partial differential equations.