Nonlinear Fokker-Planck Navier-Stokes systems. (English) Zbl 1110.35057

The author considers a fluid with microscopic inclusions. The corresponding mathematical model couples the incompressible Navier-Stokes equations to nonlinear Fokker-Planck equations. The stresses added in the fluid by the particles can depend linearly (type I) or quadratically (type II) on the density of the particles. In each case the author derives a relation for the coefficients of the stresses using energy considerations. Finally, global existence of smooth solutions to type II equations is proved provided the fluid motion is governed by Stokes equations.


35Q30 Navier-Stokes equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
76A05 Non-Newtonian fluids
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