Remarks on a Smoluchowski equation

Abstract

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.

Inertial manifolds for a Smoluchowski equation on a circle

“…However, it is widely accepted that it affords sufficient degrees of freedom to capture the dynamics of the micromicro interaction. In a recent development, the bifurcation diagram was confirmed rigorously for both the 2D and the 3D cases (see [5,6,8,13,18,19]). In the 2D case, as the potential intensity increases, the equation undergoes a pitchfork bifurcation, in which two equivalent nematic steady states (probability distribution concentrates to one direction) emerge from the isotropic one.…”

“…where m(ϕ) = (cos ϕ, sin ϕ), f ψ = 2π 0 f (ϕ)ψ(ϕ) dϕ, and the parameter b > 0 denotes the potential intensity. Regarding the existence, uniqueness and regularity of solutions of (2.1), it is easy to prove the following theorem (see [5,6]). The Smoluchowski equation preserves certain symmetries.…”

On the Doi-Onsager Model of Rigid Rod-Like Polymers

“…We begin by stating results about existence, uniqueness, positivity and regularity of the solutions of (1.1). [Constantin et al (2004)] poses the following general existence theorem without proof. According to [Zhang and Zhang (2007)], the proof can be performed using successive approximation techniques.…”

Smoluchowski Navier-Stokes systems

“…the sign of y 1 does not change during the time evolution. It was proved in ( [5]) that if the y 1 -s of two solutions converge to each other, then the solutions converge to each other. The dissipation of eneregy and the uniform lower bound on the energy can be used to show that every solution converges ultimately to a steady solution.…”