Existence of global weak solutions to the kinetic Peterlin model

We consider a class of kinetic models for polymeric fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law for an infinitely extensible spring. The polymer molecules are suspended in an incompressible viscous Newtonian fluid confined to a bounded domain in two or three space dimensions. The unsteady motion of the solvent is described by the incompressible Navier-Stokes equations with the elastic extra stress tensor appearing as a forcing term in the momentum equation. The elastic stress tensor is defined by Kramer's expression through the probability density function that satisfies the corresponding Fokker-Planck equation. In this case a coefficient depending on the average length of polymer molecules appears in the latter equation. Following the recent work of Barrett and Süli (2018) we prove the existence of global-in-time weak solutions to the kinetic Peterlin model in two space dimensions.