Existence of Large-Data Global Weak Solutions to Kinetic Models of Nonhomogeneous Dilute Polymeric Fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $\mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.