Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers

TL;DR

This paper proves the existence of global weak solutions for a class of polymeric fluid models coupling Navier-Stokes and Fokker-Planck equations, including decay to equilibrium, without structural restrictions on the drag term.

Contribution

It establishes the existence and exponential decay of global weak solutions for FENE-type bead-spring chain models with minimal assumptions on the drag term.

Findings

Existence of global-in-time weak solutions for the coupled system.

Solutions decay exponentially to equilibrium in the absence of external forces.

No structural assumptions needed on the drag term in the Fokker-Planck equation.

Abstract

We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for 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 center-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 square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.