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

## TL;DR

This paper establishes local well-posedness, a refined blow-up criterion based on density bounds, and weak-strong uniqueness for the two-dimensional compressible Oldroyd-B model, advancing understanding of solution regularity and uniqueness conditions.

## Contribution

It provides the first local well-posedness result, a refined blow-up criterion involving only density bounds, and demonstrates weak-strong uniqueness under certain initial conditions for the model.

## Key findings

- Local well-posedness in 2D for the model.
- Refined blow-up criterion based on upper density bounds.
- Weak solution coincides with strong solution under positivity and boundedness conditions.

## Abstract

We consider the compressible Oldroyd-B model derived in \cite{Barrett-Lu-Suli}, 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.