Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

Abstract

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 [Formula: see text] in [Formula: see text], [Formula: see text] or [Formula: see text], for the density [Formula: see text], the velocity [Formula: see text] and the pressure [Formula: see text] of the fluid, with an equation of state of the form [Formula: see text], where [Formula: see text] is a positive constant and [Formula: see text]. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. 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 center-of-mass diffusion term.