Global Existence for the Compressible Viscoelastic Flows with Zero Shear Viscosity and General Pressure Law in Three Dimensions

In this paper, we study the three dimensional compressible viscoelastic flows with zero shear viscosity, and a general class of pressure laws. We do not need the monotonically increasing pressure law with the help of the elasticity coefficient θ of the fluid, only need the condition P ^′(1)+θ>0. We shall reformulate the systems with the new perturbation variables (ρ−1,u,F−[Formula presented]I) and (ρ−1,u,F−I) to deal with the compressible and incompressible parts, separately. For the compressible parts, we shall use the vector fields methods to derive the weighted energy decay. For the incompressible parts, a local energy decay will be applied to derive the weighted estimates. To overcome the difficulty of the lack of dissipation for the incompressible parts, we shall introduce “good unknowns”, and use the implicit structure of the nonlinearities. With the help of vector fields, we derive the weighted L ² energy to prove global stability around a constant equilibrium.