Abstract:
The lecture will survey recent developments concerning the existence
of global-in-time 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, a crucial feature
of which is the presence of a centre-of-mass diffusion term. The proof of
the existence of global weak solutions relies on an entropy estimate
and various weak compactness techniques.
The lecture is based on a series of recent papers with John W. Barrett
(Imperial College London). |