  We consider the non-isentropic compressible Navier-Stokes equations in two or
three space dimensions for which the heat conduction of Fourier's law is
replaced by Cattaneo's law and the classical Newtonian flow is replaced by a
revised Maxwell flow. We show that a physical entropy exists for this new
model. For two special cases, we show the global well-posedness of solutions
with small initial data and the blow-up of solutions in finite time for a class
of large initial data. Moreover, for vanishing relaxation parameters, the
solutions (if it exists) are shown to converge to solutions of the classical
system.
