The talk will survey recent developments concerning the existence and the approximation of global weak solutions to a general class of coupled microscopic-macroscopic bead-spring chain models that arise in 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 for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of 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. Models of this kind were proposed in work of Hans Kramers in the early 1940's, and the existence of global weak solutions to the model has been a long-standing question in the mathematical analysis of kinetic models of dilute polymers.
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We also discuss computational challenges associated with the numerical approximation of the high-dimensional Fokker-Planck equation featuring in the model.