Existence and Equilibration of Global Weak Solutions to Hookean- Type Bead Spring Chain Models for Dilute Polymers

We show the existence of global-in-time weak solutions to a general class of coupledHookean-type bead-spring chain models that arise from the kinetic theory of dilute solu-tions of polymeric liquids with noninteracting polymer chains. The class of models involvesthe unsteady incompressible Navier–Stokes equations in a bounded domain in R, d = 2or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor ap-pearing on the right-hand side in the momentum equation. The extra-stress tensor stemsfrom the random movement of the polymer chains and is defined by the Kramers expres-sion through the associated probability density function that satisfies a Fokker–Planck-typeparabolic equation, a crucial feature of which is the presence of a center-of-mass diffusionterm. We require no structural assumptions on the drag term in the Fokker–Planck equa-tion; in particular, the drag term need not be corotational. With a square-integrable anddivergence-free initial velocity datum u∼0 for the Navier–Stokes equation and a nonnegativeinitial probability density function ψ0 for the Fokker–Planck equation, which has finite rel-ative entropy with respect to the Maxwellian M , we prove the existence of a global-in-timeweak solution t 7→ (u∼(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfyingthe initial condition (u∼(0), ψ(0)) = (u∼0, ψ0), such that t 7→ u∼(t) belongs to the classical Lerayspace and t 7→ ψ(t) has bounded relative entropy with respect to M and t 7→ ψ(t)/M hasintegrable Fisher information (w.r.t. the measure dν := M(q∼) dq∼ dx∼) over any time interval[0, T ], T > 0. If the density of body forces f∼on the right-hand side of the Navier–Stokesmomentum equation vanishes, then t 7→ (u∼(t), ψ(t)) decays exponentially in time to (0∼,M) inthe L∼×L norm, at a rate that is independent of (u∼0, ψ0) and of the centre-of-mass diffusioncoefficient.An abbreviated version of this paper has been submitted for publication in Mathematical Modelsand Methods in Applied Sciences (M3AS).