Title: Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids,
Abstract: We explore the existence of global weak solutions to the Hookean dumbbell
model, a system of nonlinear partial differential equations that arises from
the kinetic theory of dilute polymers, involving the unsteady incompressible
Navier--Stokes equations in a bounded domain in two or three space dimensions,
coupled to a Fokker--Planck-type parabolic equation. We prove the existence of
large-data global weak solutions in the case of two space dimensions.
Indirectly, our proof also rigorously demonstrates that, in two space
dimensions at least, the Oldroyd-B model is the macroscopic closure of the
Hookean dumbbell model. In three space dimensions, we prove the existence of
large-data global weak subsolutions to the model, which are weak solutions with
a defect measure, where the defect measure appearing in the Navier--Stokes
momentum equation is the divergence of a symmetric positive semidefinite
matrix-valued Radon measure.