Remarks on a Smoluchowski equation

We study the long time dynamics of aSmoluchowski equation arising in the modeling of nematic liquidcrystalline polymers. We prove uniform bounds for the long timeaverage of gradients of the distribution function in terms of thenondimensional parameter characterizing the intensity of thepotential. In the two dimensional case we obtain lower and upperbounds for the number of steady states. We prove that the systemis dissipative and that the potential serves as uniquedetermining mode of the system.