On a compressible Oldroy-B model: relative entropy, weak-strong uniqueness and conditional regularity

We consider the compressible Oldroyd-B model derived in \cite{Barrett-Lu-Suli16}, where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting. Still in the two dimensional setting, we give a (refined) blow-up criterion involving only the upper bound of the fluid density. We then show that, if the initial fluid density and polymer number density admit a positive lower bound, the weak solution coincides with the strong one as long as the latter exists. Moreover, if the fluid density of a weak solution issued from regular initial data admits a finite upper bound, this weak solution is indeed a strong one; this can be seen as a corollary of the refined blow-up criterion and the weak-strong uniqueness.