Global existence of large data weak solutions for a simplified compressible Oldroyd--B model without stress diffusion

Abstract

We start with the compressible Oldroyd--B model derived in \cite{Barrett-Lu-Suli} ({\em J. W. Barrett, Y. Lu, E. S\"uli. Existence of large-data finite-energy global weak solutions to a compressible Oldroyd--B model. Comm. Math. Sci. 15 (2017), 1265--1323}), where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting with stress diffusion. In the paper, we investigate the case without stress diffusion. We first restrict ourselves to the corotational setting as in \cite{LM00} ({\em P. L. Lions, N. Masmoudi. Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B 21(2) (2000), 131--146.}) We further assume the extra stress tensor is a scalar matrix and we derive a simplified model which takes a similar form as the multi-component compressible Navier--Stokes equations, where, however, the pressure term related to the scalar extra stress tensor has the opposite sign. By employing the techniques developed in \cite{VWY19, Novotny-Pokorny-19}, we can still prove the global-in-time existence of finite energy weak solutions in two or three dimensions, without the presence of stress diffusion.