Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids

The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of a compressible non-Newtonian fluid. In the case of the potential stress tensor, we develop a technique for deriving energy identities that do not contain derivatives of density. On the basis of these identities, in the case of sufficiently rapidly increasing potentials, we obtain an extended system ofa priori estimates for the equations mentioned above. We also study the related problem of estimating solutions to the nonlinear elliptic system generated by the stress tensor. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 360–376, September, 2000.