On a free boundary problem for finitely extensible bead-spring chain molecules in dilute polymers

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. Barrett and S\"{u}li [4], [5], [9]). 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 [19].