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# Communications in Mathematical Sciences

## Volume 6 (2008)

### Number 4

### Finite-dimensional description of the long-term dynamics for the 2D Doi-Hess model for liquid crystalline polymers in shear flow

Pages: 975 – 993

DOI: http://dx.doi.org/10.4310/CMS.2008.v6.n4.a9

#### Author

#### Abstract

The existence of inertial manifolds for a Smoluchowski equation arising in the 2D Doi-Hess model for liquid crystalline polymers subjected to a shear flow is investigated. The presence of a non-variational drift term dramatically complicates the long-term dynamics from the variational gradient case, in which it is solely characterized by the steady states. Several transformations are used in order to transform the equation into a form suitable for application of the standard theory of inertial manifolds. A nonlinear and nonlocal transformation developed in *Inertial manifolds for a Smoluchowski equation on a circle * and *Inertial manifolds for a Smoluchowski equation on the unit sphere,*, to appear, is used to eliminate the first-order derivative from the micro-micro interaction term. A traveling wave transformation eliminates the first-order derivative from the non-variational term, transforming the equation into a nonautonomous one for which the theory of nonautonomous inertial manifolds applies.

#### Keywords

Doi-Hess model; Smoluchowski equation; shear flow; nonautonomous inertial manifolds; Schrödinger-like equation

#### 2010 Mathematics Subject Classification

35Kxx, 70Kxx