Dynamics of concentrated polymer solutions are modelled by a Smoluchowski equation. At high concentrations, such solutions form liquid crystalline polymers of nematic structure. We prove that at high intensities the twodimensional Smoluchowski equation possesses exactly two steady states, corresponding to the isotropic and the nematic phases.

We investigate a Smoluchowski equation (a nonlinear FokkerPlanck equation on the unit sphere), which arises in modeling of colloidal suspensions. We prove the dissipativity of the equation in 2D and 3D, in certain Gevrey classes of analytic functions.

The existence of inertial manifolds for a Smoluchowski equation-a nonlinear and nonlocal Fokker-Planck equation which arises in the modelling of colloidal suspensions-is investigated. The difficulty due to first-order derivatives in the nonlinearity is circumvented by a transformation.

The existence of inertial manifolds for a Smoluchowski equationa nonlinear Fokker-Planck equation on the unit sphere which arises in modeling of colloidal suspensions -is investigated. A nonlinear and nonlocal transformation is used to eliminate the gradient from the nonlinear term.

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to either Dirichlet or von Neumann boundary conditions and study both time-independent and time-periodic cases involving Liouville integrable Hamiltonians that satisfy conditions conducive to applying the averaging principle. Transformation to action-angle coordinates permits averaging in time and angle, leading to an underlying eigenvalue equation that allows for separation of the angle and action coordinates. The result is a one-dimensional second-order equation involving an anti-symmetric imaginary potential. For radial flows on a disk or an annulus, we rigorously apply existing complex-plane WKBJ methods to study the spectral properties in the semi-classical limit for vanishing diffusivity. In this limit, the spectrum is found to be a complicated set consisting of lines related to Stokes graphs. Eigenvalues in the neighborhood of these graphs exhibit nonlinear scaling with respect to diffusivity leading to convection-enhanced rates of dissipation (relaxation, mixing) for initial data which are mean-free in the angle coordinate. These branches coexist with a diffusive branch of eigenvalues that scale linearly with diffusivity and contain the principal eigenvalue (no dissipation enhancement).