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# Communications in Mathematical Sciences

## Volume 8 (2010)

### Number 3

### Special Issue: Mathematical Issues of Complex Fluids

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

Pages: 763 – 782

DOI: http://dx.doi.org/10.4310/CMS.2010.v8.n3.a8

#### Authors

#### 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.

#### Keywords

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

#### 2010 Mathematics Subject Classification

35B40, 35K15, 35Q30, 76T20