McKean-Vlasov diffusion and the well-posedness of the Hookean bead-spring-chain model for dilute polymeric fluids: small-mass limit and equilibration in momentum space

Abstract: We reformulate a general class of classical bead-spring-chain models for dilute polymeric fluids, with Hookean spring potentials, as McKean-Vlasov diffusion. This results in a coupled system of partial differential equations involving the unsteady incompressible linearized Navier-Stokes equations, referred to as the Oseen system, for the velocity and the pressure of the fluid, with a source term which is a nonlinear function of the probability density function, and a second-order degenerate parabolic Fokker-Planck equation, whose transport terms depend on the velocity field, for the probability density function. We show that this coupled Oseen-Fokker-Planck system has a large-data global weak solution. We then perform a rigorous passage to the limit as the masses of the beads in the bead-spring-chain converge to zero, which is shown in particular to result in equilibration in momentum space. The limiting problem is then used to perform a rigorous derivation of the Hookean bead-spring-chain model for dilute polymeric fluids, which has the interesting feature that, if the flow domain is bounded, then so is the associated configuration space domain and the associated Kramers stress tensor is defined by integration over this bounded configuration domain.