Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity

We prove the existence of global-in-time weak solutions to a general class of coupled bead–spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains, with finitely extensible nonlinear elastic (FENE) spring potentials. The class of models under consideration involves the unsteady incompressible Navier–Stokes equations with variable density and density-dependent dynamic viscosity in a bounded domain in R^d, d=2 or 3, for the density, the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term and a nonlinear density-dependent drag coefficient. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With initial density ρ_0∈[ρ_min,ρ_max] for the continuity equation, where ρ_min>0; a square-integrable and divergence-free initial velocity datum for the Navier–Stokes equation; and a nonnegative initial probability density function ψ_0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition such that t↦ρ(t)∈[ρ_min,ρ_max], belongs to the classical Leray space and t↦ψ(t) has bounded relative entropy with respect to M and t↦ψ(t)/M has integrable Fisher information (w.r.t. the Gibbs measure) over any time interval [0,T], T>0.