Smoluchowski Navier-Stokes Systems

We discuss equilibria, dynamics and regularity for Smoluchowski equations coupled to Navier-Stokes equations. Introduction We consider mixtures of fluids and microscopic inclusions. The microscopic inclusions are characterized by state variables m ∈ M , where M is a compact smooth Riemannian manifold without boundary. The simplest example is that of microscopic rods with directors m ∈ S. The microscopic inclusions evolve stochastically: they are carried by the ambient fluid, agitated by thermal noise and interact with each other. This behavior is modeled in this paper by a Smoluchowski equation for the probability distribution of particles. The inclusions add stresses to the fluid, and thus the system is coupled. When the coupling is negligible, and the inclusions are in statistical equilibrium, then the system is governed by a single time-independent equation, derived by Onsager for colloidal suspensions of rod-like particles. This equation is variational in nature, nonlinear and nonlocal. The free energy has an entropic part and a microscopic selfinteraction part. The selfinteractions are quadratic but nonlocal and indefinite. In the particular case of a specific microscopic model given by a Maier-Saupe potential, Onsager’s equations reduce to few transcendental implicit equations. These can be analyzed, and the limit of strong microscopic interactions can be shown to have a nematic character, which means in this context that the probability distributions concentrate to singular sets in M . High intensity asymptotics for Onsager’s equations have been studied in ([4]). Qualitative properties of solutions were obtained in ([11], [21], [22], [12]). The transition to nematic states as the intensity of the selfinteractions increases, as well as the fact that the infinite dimensional nonlinear nonlocal Onsager equation reduces to few transcendental equations with a variational structure are not isolated features, due to the fact that the Maier-Saupe interactions are particularly limited. In fact, for generic interactions, Onsager’s equation can be written as a sequence of transcendental equations, and the high interaction limit is generically a delta function on M . 1991 Mathematics Subject Classification. Primary 35Q30, 82C31; Secondary 76A05.