We consider the existence of global-in-time weak solutions in two spatial dimensions to the Hookean dumbbell model, which arises as a microscopic-macroscopic bead-spring model from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. This model involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in 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 center-of-mass diffusion term. We show the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that the Oldroyd-B model is a macroscopic closure of the Hookean dumbbell model in two space dimensions. Finally, we show the existence of large-data global weak subsolutions to the Hookean dumbbell model in the case of three space dimensions.
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