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On a free boundary problem for finitely extensible bead-spring chain molecules in dilute polymers. (English) Zbl 1427.35372

Summary: 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. We construct weak solutions of the two-phase model by performing the asymptotic limit as the adiabatic exponent \(\gamma\) goes to \(\infty\) for a macroscopic model which arises from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids. In this context the polymeric molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. This class of models involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain \(\Omega\) in \(\mathbb{R}^d, d = 2, 3\) coupled with a Fokker-Planck-Smoluchowski-type diffusion equation (cf. [J. W. Barrett and E. Süli, Math. Models Methods Appl. Sci. 21, No. 6, 1211–1289 (2011; Zbl 1244.35101); ibid. 22, No. 5, 1150024, 84 p. (2012; Zbl 1237.35127); ibid. 26, No. 3, 469–568 (2016; Zbl 1336.35273)]). The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using weak convergence methods, compactness arguments which rely on the monotonicity properties of certain quantities in the spirit of [D. Donatelli and K. Trivisa, NoDEA, Nonlinear Differ. Equ. Appl. 24, No. 5, Paper No. 51, 20 p. (2017; Zbl 1382.35187)].

MSC:

35R35 Free boundary problems for PDEs
35Q30 Navier-Stokes equations
35Q84 Fokker-Planck equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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References:

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