Relative Entropy, Weak-Strong Uniqueness, and Conditional Regularity for a Compressible Oldroyd--B Model

Abstract

We consider the compressible Oldroyd--B model derived in [J. W. Barrett, Y. Lu, and E. Süli, Comm. Math. Sci., 15 (2017), pp. 1265--1323], where the existence of global-in-time finite-energy weak solutions was shown in a two-dimensional setting. In this paper, we first state a local well-posedness result for this compressible Oldroyd--B model. 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 issuing 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.