Global solutions and uniform convergence stability for compressible Navier–Stokes equations with Oldroyd-type constitutive law

We consider a class of physically relevant one-dimensional isentropic compressible Navier–Stokes equations with viscoelastic constitutive law of Oldroyd type. By establishing uniform a priori estimates (with respect to relaxation time), we show global existence of smooth solutions with small initial data. Moreover, we get global-in-time convergence of the system toward the classical isentropic compressible Navier–Stokes equations.