Existence of Global Weak Solutions to Coupled Navier–stokes–fokker–planck Systems: a Brief Survey

We present a brief survey of recent results concerning the existence of global-in-time weak solutions in a bounded Lipschitz domain in R, d ∈ {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. AMS Mathematics Subject Classification (2000): 35Q30, 76A05, 76D03, 82C31, 82D60