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We show the existence of global-in-time weak solutions to a general class ofcoupled bead-spring chain models that arise from the kinetic theory of dilutesolutions of nonhomogeneous polymeric liquids with noninteracting polymerchains, with finitely extensible nonlinear elastic (FENE) spring potentials.The class of models under consideration involves the unsteady incompressibleNavier-Stokes equations with variable density and density-dependent dynamicviscosity in a bounded domain in two and three space dimensions, for thedensity, the velocity and the pressure of the fluid, with an elasticextra-stress tensor appearing on the right-hand side in the momentum equation.The extra-stress tensor stems from the random movement of the polymer chainsand is defined by the Kramers expression through the associated probabilitydensity that satisfies a Fokker-Planck-type parabolic equation, a crucialfeature of which is the presence of a centre-of-mass diffusion term and anonlinear density-dependent drag coefficient. With a bounded and positiveinitial density for the continuity equation; a square-integrable anddivergence-free initial velocity datum for the Navier-Stokes equation; and anonnegative initial probability density function for the Fokker-Planckequation, which has finite relative entropy with respect to the Maxwellianassociated with the spring potential in the model, we prove, via a limitingprocedure on certain regularization parameters, the existence of aglobal-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system,satisfying the given initial condition.

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