Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients

Donatella Donatelli; Tessa Thorsen; Konstantina Trivisa

doi:10.3934/krm.2020037

Article Contents

2020, Volume 13, Issue 5: 1047-1070. Doi: 10.3934/krm.2020037

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Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients

1.
Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, 67100 L'Aquila, Italy
2.
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Received: January 2020

Revised: June 2020

Published: August 2020

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Abstract

We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. The free boundary in the present context is defined with regard to a density threshold of $ \rho = 1, $ below which the fluid is modeled as compressible and above which the fluid is modeled as incompressible. The present article focuses on the physically relevant case in which the viscosity coefficients present in the system depend on the polymer number density, extending the earlier work [8]. We construct the weak solutions of the free boundary problem by performing the asymptotic limit as the adiabatic exponent $ \gamma $ goes to $ \infty $ for the macroscopic model introduced by Feireisl, Lu and Süli in [10] (see also [6]). The weak sequential stability of the family of dissipative (finite energy) weak solutions to the free boundary problem is also established.

Keywords:

FENE model,
suspensions of extensible bead-spring chain molecules,
dilute polymers,
compressible Navier-Stokes equations,
Fokker-Planck-type equation,
free boundary problems.

Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 46E35.

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References

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