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 \(α ∈ ( 12,1)\) and derive an energy inequality satisfied by weak solutions.Keywordsexistence of weak solutionsNavier–Stokes–Fokker–Planck systemdilute polymer modelHookean and FENE-type bead-spring-chain modelRiemann–Liouville fractional derivativetime-fractional PDEMSC codes35Q3035Q8435R1160G2282C3182D60