Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows

Hu, Xianpeng; Wu, Guochun

doi:10.1137/120892350

Cited by 93 publications

(61 citation statements)

References 24 publications

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“…For a survey of macroscopic models of compressible viscoelastic flow, the reader is referred to the paper by Bollada & Phillips [18]. 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 [34,61,60,35,36,37]. The existence of measurevalued solutions to non-Newtonian compressible, isothermal, monopolar fluid flow models was studied by Nečasová in [53,55]; for bipolar isothermal non-Newtonian compressible fluids related analysis was pursued in [54].…”

Section: Introductionmentioning

confidence: 99%

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

Barrett

Süli

2016

Math. Models Methods Appl. Sci.

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Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Ω in R d , d = 2 or 3, for the density ρ, the velocity u ∼ and the pressure p of the fluid, with an equation of state of the form p(ρ) = cpρ γ , where cp is a positive constant and γ > 3 2. The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term.

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Section: Introductionmentioning

confidence: 99%

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

Barrett

Süli

2016

Math. Models Methods Appl. Sci.

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“…Remark In , the authors obtained the optimal convergence rates of the global strong solutions to the system when the initial data belong to

H^{2} (R_{x}^{3}) \cap L^{1} (R_{x}^{3})

. Compared with the existing result, we obtain the optimal convergence rates of the global strong solutions to the system under the lowest regularity assumptions.…”

Section: Introduction and Main Resultsmentioning

confidence: 83%

“…However, to the best of our knowledge, very few results have been established on the dynamics of the global solutions to the multi‐dimensional compressible viscoelastic flows, especially on the large time behavior. On the one hand, X. Hu and G. Wu in first gave a detailed analysis about the global existence and the optimal time‐decay rates of strong solutions to the three‐dimensional compressible viscoelastic flows where the initial data are close to an equilibrium state and belong to

H^{2} (R_{x}^{3}) \cap L^{1} (R_{x}^{3})

. Here,

L^{1} (R_{x}^{3})

integrability plays an important role in the proof of the optimal convergence rates and

H^{2} (R_{x}^{3})

has a higher regular index than the critical regular index.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The optimal convergence rates for the multi-dimensional compressible viscoelastic flows

Zhang

et al. 2016

Z. Angew. Math. Mech.

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“…Let us mention some mathematical results for compressible viscoelastic models, which has been the subject of active research in recent years. 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 [27,21,22,23]. In particular, Fang and Zi [12] proved the existence of a unique local-in-time strong solution to a compressible Oldroyd-B model and established a blow-up criterion for strong solutions.…”

Section: )mentioning

confidence: 99%

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

Lu¹,

Zhang²

2018

SIAM J. Math. Anal.

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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.

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Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows

Cited by 93 publications

References 24 publications

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

The optimal convergence rates for the multi-dimensional compressible viscoelastic flows

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

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