Coupled Navier-Stokes-Fokker-Planck Systems

John W. Barrett, Endre Süli

We present a brief survey of recent results concerning the existence of global-in-time weak solutions in a bounded Lipschitz domain in $\mathbb R^d$, $d\in\{2,3\}$, to a class of kinetic models for dilute polymeric liquids with noninteracting polymer chains. The mathematical model is a coupled Navier-Stokes-Fokker-Planck system. The velocity and the pressure of the fluid satisfy a Navier-Stokes-like system of partial differential equations, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The elastic extra-stress tensor stems from the random movement and stretching of the polymer chains and is defined through the associated probability density function, which satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term, an unbounded drift term, and microscopic cut-off in the drag term. The Fokker-Planck equation admits a general class of unbounded spring-force potentials, including in particular the FENE (Finitely Extensible Nonlinear Elastic) potential.