We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type
spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier{Stokes system in a bounded domain in Rd, d= 2, for the density p, the velocity u and the pressure
p of the uid, with an equation of state of the form p (p) = cp p, where cp
is a positive constant and >1. The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined 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. This extends the result in our paper [J.W. Barrett & E. Suli: Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci.,
26 (2016)], which established the existence of global-in-time weak solutions to the system for dE {2, 3} and > 3/2, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d = 2 and
> 1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density p0 E L$ (Ω)
for the continuity equation; a square-integrable initial velocity datum u0 for
the Navier-Stokes momentum equation; and a nonnegative initial probability density function $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 a pressure regularization parameter, the existence of a global-in-time bounded-energy weak solution
t→ (p(t), u(t), $(t) to the coupled Navier-Stokes-Fokker-Planck system, satisfying the initial condition (p(0), u(0), $(0) - (p0, u0, $0)