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

TL;DR

This paper proves the global existence of weak solutions for a coupled Navier-Stokes-Fokker-Planck system with a time-fractional derivative, modeling subdiffusive polymer solutions.

Contribution

It establishes the well-posedness and energy inequalities for a novel time-fractional polymer model derived from a subordinated Langevin equation.

Findings

Proved global-in-time existence of weak solutions for the model.

Derived an energy inequality for weak solutions.

Model captures subdiffusive behavior in polymer solutions.

Abstract

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 modelled 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 modelled by a finitely extensible nonlinear elastic (FENE) 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 $\alpha \in (\tfrac12,1)$, and derive an energy inequality satisfied by weak solutions.