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

Abstract

We consider the compressible Oldroyd-B model derived in [2], where the existence of globalin-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.

Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

“…Dostalík, Průša and Stein ( [16]) studied the large time behaviors for the related viscoelastic models with the free energy estimates in three-dimensional vessel. For compressible case, please refer to [2,42,49] for global existence of weak solutions, weak-strong uniqueness and conditional regularity, and optimal time-decay estimates, respectively.…”

“…As showed by Málek et al ([43]), the stress diffusion term can be interpreted either as a consequence of a nonlocal energy storage mechanism or as a consequence of a nonlocal entropy production mechanism. Thus the diffusive Oldroyd-B system has attracted much attention and been studied extensively, see [1,2,12,13,14,18,19,33,42,49].…”

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

“…Here we shall employ the well-known relative entropy inequality to derive the basic energy estimate for ̺ ε and R ε . Relative entropy inequality is widely used in the study of uniqueness and stability for Navier-Stokes equations and some related models, see [7,10] for compressible Navier-Stokes equations, [8] for compressible Navier-Stokes-Fourier equations, and [18] for a compressible Oldroyd model. The result states as follows: Proposition 2.5.…”

Global Existence of Large Data Weak Solutions for a Simplified Compressible Oldroyd--B Model Without Stress Diffusion

“…In [2], not only the compressible Oldroyd-B model (1.1a)-(1.3) was derived, but also the existence of global-in-time weak solutions in two dimensional setting was shown. Recently in [29], one of the authors and his collaborator proved the weak-strong uniqueness, gave a refined blow-up criterion and showed a conditional regularity result in two dimensional setting.…”