Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers

TL;DR

This paper proves the existence of global weak solutions for a class of coupled Navier-Stokes and Fokker-Planck models describing dilute polymer solutions, showing exponential decay to equilibrium without structural assumptions on the drag term.

Contribution

It establishes the existence of global-in-time weak solutions for a broad class of Hookean-type bead-spring models without requiring structural assumptions on the drag term.

Findings

Existence of global weak solutions for the coupled system.

Solutions decay exponentially to equilibrium.

No structural assumptions needed on the drag term.

Abstract

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-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.