Remarks on a Smoluchowski equation

Peter Constantin; Ioannis Kevrekidis; E. S. Titi

doi:10.3934/dcds.2004.11.101

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2004, Volume 11, Issue 1: 101-112. Doi: 10.3934/dcds.2004.11.101

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Remarks on a Smoluchowski equation

1.
Department of Mathematics, The University of Chicago, Chicago, Il 60637
2.
Chemical Engineering, PACM and Mathematics, Princeton University, Princeton, NJ 08544
3.
Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697

Received: October 31, 2002

Revised: September 30, 2003

Published: December 2003

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Abstract

We study the long time dynamics of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. We prove uniform bounds for the long time average of gradients of the distribution function in terms of the nondimensional parameter characterizing the intensity of the potential. In the two dimensional case we obtain lower and upper bounds for the number of steady states. We prove that the system is dissipative and that the potential serves as unique determining mode of the system.

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Mathematics Subject Classification: 35K35, 76A05, 76A15, 82D60.

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