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January  2004, 11(1): 101-112. doi: 10.3934/dcds.2004.11.101

Remarks on a Smoluchowski equation

Peter Constantin 1, , Ioannis Kevrekidis 2, and  E. S. Titi 3,

1. 

Department of Mathematics, The University of Chicago, Chicago, Il 60637
2. 

Chemical Engineering, PACM and Mathematics, Princeton University, Princeton, NJ 08544, United States
3. 

Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States

Received  November 2002 Revised  October 2003 Published  April 2004

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.
Keywords: gradient bounds, Rheology, nematic liquid crystalline polymers, determining modes., Smoluchowski equation, number of steady solutions, non-Newtonian fluids, long time dynamics.
Mathematics Subject Classification: 35K35, 76A05, 76A15, 82D6.
Citation: Peter Constantin, Ioannis Kevrekidis, E. S. Titi. Remarks on a Smoluchowski equation. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 101-112. doi: 10.3934/dcds.2004.11.101
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