Abstract This paper is concerned with the Cauchy problem of the 2D compressible viscous Oldroyd-B model with stress diffusion. The global well-posedness of strong solutions with large data is established, provided the bulk viscosity depends on the density in the specific form λ ( ρ ) = ρ β with β > 4 / 3 and the initial data decay suitably fast at infinity. The spatially asymptotic behaviour of the density, the momentum, the polymer number density and the extra stress tensor are also obtained. It is worth mentioning that the spatial decay properties of higher-order derivatives of the solutions are entirely new, revealing previously unknown asymptotic structures even for the compressible Navier–Stokes equations (cp. Huang and Li (2022 SIAM J. Math. Anal. 54 3192–241)). A key observation is that the trace of extra stress tensor remains nonnegative if it starts nonnegative.
Liu et al. (Tue,) studied this question.