Global Existence of Solutions of the Compressible Viscoelastic Fluid Around a Parallel Flow

We consider a system of equations for a motion of a compressible viscoelastic fluid in an infinite layer. When the external force has a suitable form, the system has a solution of parallel flow type. It is shown that the solution of the system exists globally in time if the initial data is sufficiently close to the one of the parallel flow, provided that the Reynolds number and the Mach number are sufficiently small, and that the speed of the propagation of shear wave is sufficiently large.