Analysis of a dilute polymer model with a time-fractional derivative

We investigate the well-posedness of a coupled Navier-Stokes-Fokker-Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modeled by a stochastic process exhibiting power-law waiting time in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modeled by a finitely extensible nonlinear elastic dumbbell model, and the drag term in the Fokker-Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order alpha is an element of (1/2,1) and derive an energy inequality satisfied by weak solutions.