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

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 $\alpha \in (\frac{1}{2},1)$, and derive an energy inequality satisfied by weak solutions.

References in corpus (13)

• Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
• Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers
• Numerical solution of the time-fractional Fokker-Planck equation with general forcing
• Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers
• Regularity theory for time-fractional advection-diffusion-reaction equations
• Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing
• On a subdiffusive tumour growth model with fractional time derivative
• Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
• Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
• Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials
• On diffusive 2D Fokker-Planck-Navier-Stokes systems
• The FENE dumbbell polymer model: existence and uniqueness of solutions for the momentum balance equation
• A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data

Cited by in corpus (1)

• Numerical simulation of dilute polymeric fluids with memory effects in the turbulent flow regime