Existence of global weak solutions to compressible isentropic finitely extensible nonlinear 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 $\Omega$ in $\mathbb{R}^d$, $d = 2$ or $3$, for the density, the velocity and the pressure of the fluid. 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 centre-of-mass diffusion term. We 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 for the continuity equation; a square-integrable initial velocity datum for the Navier-Stokes momentum equation; and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian associated with the spring potential in the model, we prove, via a limiting procedure on certain discretization and regularization parameters, the existence of a global-in-time bounded-energy weak solution to the coupled Navier-Stokes-Fokker-Planck system, satisfying the prescribed initial condition.