## Communications in Mathematical Sciences

- Commun. Math. Sci.
- Volume 8, Number 3 (2010), 763-782.

### Refined long-time asymptotics for some polymeric fluid flow models

A. Arnold, J. A. Carrillo, and C. Manzini

#### Abstract

We consider a polymeric fluid model, consisting of the incompressible Navier-Stokes equations coupled to a non-symmetric Fokker-Planck equation. First, the existence of steady states and the exponential convergence to them in relative entropy are proved for the linear Fokker-Planck equation in the Hookean case. The FENE model is also addressed, and the proof of the existence of stationary states and the convergence towards them in suitable weighted norms is given. Then, using the “entropy method” exponential convergence to the steady state is established for the coupled model in the Hookean case under some smallness assumption. The results continue and expand the analysis of B. Jourdain, C. Le Bris, T. Lelièvre and F. Otto, Arch. Rational Mech. Anal., 181, 97-148, 2006 in both the Hookean and the FENE models.

#### Article information

**Source**

Commun. Math. Sci., Volume 8, Number 3 (2010), 763-782.

**Dates**

First available in Project Euclid: 25 August 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.cms/1282747139

**Mathematical Reviews number (MathSciNet)**

MR2730330

**Zentralblatt MATH identifier**

1213.35094

**Subjects**

Primary: 35K15: Initial value problems for second-order parabolic equations 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35B40: Asymptotic behavior of solutions 76T20: Suspensions

**Keywords**

Entropy method relative entropy Fokker-Planck equations large time behavior exponential decay rate polymeric flow dumbbell model

#### Citation

Arnold, A.; Carrillo, J. A.; Manzini, C. Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8 (2010), no. 3, 763--782. https://projecteuclid.org/euclid.cms/1282747139