American Institute of Mathematical Sciences

October  2020, 13(5): 1047-1070. doi: 10.3934/krm.2020037

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

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.

Citation: Donatella Donatelli, Tessa Thorsen, Konstantina Trivisa. Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients. Kinetic and Related Models, 2020, 13 (5) : 1047-1070. doi: 10.3934/krm.2020037
References:
 [1] H. Bae and K. Trivisa, On the Doi model for the suspensions of rod-like molecules in compressible fluids, Math. Models Methods Appl. Sci., 22 (2012), 39 pp. doi: 10.1142/S0218202512500273. [2] H. Bae and K. Trivisa, On the Doi model for the suspensions of rod-like molecules: Global-in-time existence, Commun. Math. Sci., 11 (2013), 831-850.  doi: 10.4310/CMS.2013.v11.n3.a8. [3] J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅰ: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21 (2011), 1211-1289.  doi: 10.1142/S0218202511005313. [4] J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅱ: Hookean-type bead-spring chains, Math. Models Methods Appl. Sci., 22 (2012), 84 pp. doi: 10.1142/S0218202511500242. [5] J. W. Barrett and E. Süli, Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity, preprint, arXiv: 1112.4781. doi: 10.1016/j.jde.2012.09.005. [6] J. W. Barrett and E. Süli, Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci., 26 (2016), 469-568.  doi: 10.1142/S0218202516500093. [7] D. Donatelli and K. Trivisa, On a free boundary problem for polymeric fluids: Global existence of weak solutions, NoDEA Nonlinear Differential Equations Appl., 4 (2017), 20 pp. doi: 10.1007/s00030-017-0475-5. [8] D. Donatelli and K. Trivisa, On a free boundary problem for finitely extensible bead-spring chain molecules in dilute polymers, J. Math. Anal. Appl., 482 (2020), 24 pp. doi: 10.1016/j.jmaa.2019.123527. [9] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98. [10] E. Feireisl, Y. Lu and E. Süli, Dissipative weak solutions to compressible Navier-Stokes-Fokker-Planck systems with variable viscosity coefficients, J. Math. Anal. App., 443 (2016). doi: 10.1016/j.jmaa.2016.05.030. [11] P.-L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. Henri Poincaré, 16 (1999), 373-410.  doi: 10.1016/S0294-1449(99)80018-3.

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References:
 [1] H. Bae and K. Trivisa, On the Doi model for the suspensions of rod-like molecules in compressible fluids, Math. Models Methods Appl. Sci., 22 (2012), 39 pp. doi: 10.1142/S0218202512500273. [2] H. Bae and K. Trivisa, On the Doi model for the suspensions of rod-like molecules: Global-in-time existence, Commun. Math. Sci., 11 (2013), 831-850.  doi: 10.4310/CMS.2013.v11.n3.a8. [3] J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅰ: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21 (2011), 1211-1289.  doi: 10.1142/S0218202511005313. [4] J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅱ: Hookean-type bead-spring chains, Math. Models Methods Appl. Sci., 22 (2012), 84 pp. doi: 10.1142/S0218202511500242. [5] J. W. Barrett and E. Süli, Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity, preprint, arXiv: 1112.4781. doi: 10.1016/j.jde.2012.09.005. [6] J. W. Barrett and E. Süli, Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci., 26 (2016), 469-568.  doi: 10.1142/S0218202516500093. [7] D. Donatelli and K. Trivisa, On a free boundary problem for polymeric fluids: Global existence of weak solutions, NoDEA Nonlinear Differential Equations Appl., 4 (2017), 20 pp. doi: 10.1007/s00030-017-0475-5. [8] D. Donatelli and K. Trivisa, On a free boundary problem for finitely extensible bead-spring chain molecules in dilute polymers, J. Math. Anal. Appl., 482 (2020), 24 pp. doi: 10.1016/j.jmaa.2019.123527. [9] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98. [10] E. Feireisl, Y. Lu and E. Süli, Dissipative weak solutions to compressible Navier-Stokes-Fokker-Planck systems with variable viscosity coefficients, J. Math. Anal. App., 443 (2016). doi: 10.1016/j.jmaa.2016.05.030. [11] P.-L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. Henri Poincaré, 16 (1999), 373-410.  doi: 10.1016/S0294-1449(99)80018-3.
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