The Liouville theorem and the $L^2$ decay of the FENE dumbbell model of polymeric flows

TL;DR

This paper proves a Liouville theorem for the steady-state FENE model, showing only trivial solutions under certain conditions, and establishes decay rates for the velocity in the co-rotation FENE model across different dimensions.

Contribution

It generalizes classical results to the FENE model, proves a Liouville theorem, and determines decay rates for the velocity in the FENE model, improving previous results.

Findings

Trivial solutions for steady-state FENE model under integrability conditions.

Velocity decay rate of (1+t)^{-d/4} for d≥3.

Logarithmic decay rate for d=2.

Abstract

In this paper we mainly investigate the finite extensible nonlinear elastic (FENE) dumbbell model with dimension $d\geq2$ in the whole space. We first proved that there is only the trivial solution for the steady-state FENE model under some integrable condition. Our obtained results generalize and cover the classical results to the stationary Navier-Stokes equations. Then, we study about the $L^2$ decay of the co-rotation FENE model. Concretely, the $L^2$ decay rate of the velocity is $(1+t)^{-\frac{d}{4}}$ when $d\geq3$, and $\ln^{-k}{(e+t)}, k\in \mathds{N}^{+}$ when $d=2$. This result improves considerably the recent result of \cite{Schonbek2} by Schonbek. Moreover, the decay of general FENE model has been considered.