Global convergence rates from relaxed Euler equations to Navier-Stokes equations with Oldroyd-type constitutive laws

In a previous work (Peng and Zhao 2022 J. Math. Fluid Mech. 24 29), it is proved that the 1D full compressible Navier-Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd's derivatives and a revised Cattaneo's constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier-Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.