The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids

Abstract : The initial value problem associated with the equations of motion for isotropic Newtonian fluids is investigated. The fluids are compressible, viscous and heat-conductive. It is proved that there exists a unique global solution in time, for the small initial data, and the solution has the decay rate of (1 + t) to 3/4 power as t approaches positive infinity. The motions of compressible, viscous and heat-conductive fluids are described by a system of partial differential equations which is of hyperbolic-parabolic type and highly nonlinear. One of the first mathematical problems associated with this system is the initial value problem. We obtain the existence of a a unique smooth global solution in time for the initial value problem and also the decay rate of the solution as time tends to infinity.