The role of the free work identity in viscous compressible flows

Aims: (1) study of nonlinear stability in the mean of rest state of viscous or inviscid fluids; (2) study of nonlinear asyptotical decay of perturbations to rest state} of viscous fluids. Tools: (1) conservation laws for the construction of a Ljapunov functional; (2)] suitable choice of test functions in the weak formulation of the problem. Results: (I) it is stable any rest state that is a proper minimum for E where E is the total energy, S the entropy; (II) any dissipative system (having V( u) not 0, k is asyptotically converging to the rest state. Here we consider conservative external forces in order to have only rest state as steady motion.

The free work equation represents a direct method for the study of nonlinear stability of a mixed hyperbolic-parabolic system. We apply such a method to study the stability of a steady solution of a viscous barotropic fluid.