Special Session 7: 

Relative entropy, weak-strong uniqueness and conditional regularity for a compressible Oldroyd--B model

Yong Lu
Nanjing University
Peoples Rep of China
Co-Author(s):    Zhifei Zhang
We consider the compressible Oldroyd--B model derived in the paper {\em Existence of large-data finite-energy global weak solutions to a compressible Oldroyd--B model}, Comm. Math. Sci. 15 (2017), 1265--1323 by J. W. Barrett, Y. Lu and E. S\"uli, where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting. In this paper, we first state a local well-posedness result for this compressible Oldroyd--B model. In 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.