Decay of the kinetic and the thermal energy of compressible viscous fluids

We consider a heat conducting compressible viscous fluid at rest in a rigid container and show that the thermal and the kinetic energy of arbitrary disturbances of the rest state eventually decays. REsuME. On considere un fluide compressible, visqueux et conducteur de la chaleur au repos dans un reservoir rigide. On montre que l'energie thermique et cinetique de perturbations arbitraires est amortie. . Introduction In [lJ, [2J I extended Kampe de Feriet's [3J result concerning the decay of the energy of incompressible viscous fluids filling a closed rigid container to heat conducting incompressible viscous fluids and also to the case when the fluid does not fill the vessel. Here I study the corresponding problem for heat conducting compressible Navier-Stokes fluids and show that the thermal and the kinetic energy of arbitrary disturbances of the rest state eventually decays. It would seem that for the mechanical problem, one should be able to obtain decay rate of the kinetic energy from the analysis of Shahinpoor and Ahmadi [4J. However, these authors make an assumption which for Navier-Stokes fluids implies that after a certain time, the rate at which body forces do work balances the rate of work done by the hydrostatic pressure. I do not make any such assumption and obtain a slightly weaker result. * Department of Engineering Mechanics, University of Missouri-Rolla, Rolla, Missouri 65401, USA.