Strong solutions of 3D compressible Oldroyd‐B fluids

Abstract

This paper is concerned with a compressible viscoelastic fluids of Oldroyd‐B type. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. Moreover, we establish a blow‐up criterion for the strong solution in terms of the Lt∞Lx∞ norm of the density tensor ρ and the Lt2Lx∞ norm of the symmetric tensor of constraints τ. All the results hold for the initial density vanishing from below. Copyright © 2012 John Wiley & Sons, Ltd.

Stress-diffusive regularizations of non-dissipative rate-type materials

“…He also studied the incompressible limit problem and showed that compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero. Strong solutions of three-dimensional flows of compressible Oldroyd-B fluids were studied in Fung and Zi [14]. Recently, Barrett et al [2] established long-time and large-data existence of weak solutions to compressible Oldroyd-B fluids with stress diffusion.…”

“…A remarkable difference between (13) and (14) is that the termṠ + SW − WS in (14) is objective whilė S in (13) is not.…”

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

“…Theorem 2.3 is given in a similar manner as Theorem 1.1 and Theorem 1.2 in [12] and can be proved similarly. In fact, there are extra diffusion terms in our model compared to the model considered in [12], and this makes the proof even easier. Hence, we omit the proof of this theorem.…”

“…In two dimensional setting, we offer the following refined blow-up criterion: Remark 2.7. The blow-up criterion (2.12), which is reproduced from [12], is inspired by the related study for compressible Navier-Stokes equations in [30,31] and for incompressible Oldroyd-B model in [9]. Our refined one in Theorem 2.6 coincides with those in [30,31] where only the upper bound of the fluid density is needed.…”

Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model

“…The existence and uniqueness of local strong solutions and the existence of global solutions near equilibrium for macroscopic models of three-dimensional compressible viscoelastic fluids was considered in [27,45,44,28,29,30]. Fang and Zi [17] proved the existence of a unique local strong solution to a compressible Oldroyd-B model for all initial data satisfying a certain compatibility condition, and established a blow-up criterion for strong solutions. Lei [33] proved the local and global existence of classical solutions to a compressible Oldroyd-B system in a torus with small initial data; he also studied the incompressible limit problem and showed that solutions to the compressible flow model with well-prepared initial data converge to those of the incompressible model when the Mach number converges to zero.…”