Nonlinear Fokker-Planck Navier-Stokes systems

Abstract

We consider Navier-Stokes equations coupled to nonlinear Fokker-Planck equations describing the probability distribution of particles interacting with fluids. We describe relations determining the coefficients of the stresses added in the fluid by the particles. These relations link the added stresses to the kinematic effect of the fluid's velocity on particles and to the inter- particle interaction potential. In equations of type I, where the added stresses depend linearly on the particle distribution density, energy balance requires a response potential. In equations of type II, where the added stresses depend quadratically on the particle distribution, energy balance can be achieved without a dynamic response potential. In unforced energetically balanced equations, all the steady solutions have fluid at rest and particle distributions obeying an uncoupled Onsager equation. Systems of equations of type II have global smooth solutions if inertia is neglected.