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

Abstract 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.