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

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

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Published 1 January 2008