TY - JOUR

T1 - NONLINEAR FOKKER-PLANCK NAVIER-STOKES SYSTEMS

AU - Constantin, Peter

N1 - Funding Information:
Research partially supported by NSF-DMS grant 0504213
Funding Information:
Research partially supported by NSF-DMS grant
Publisher Copyright:
© 2005 International Press

PY - 2005

Y1 - 2005

N2 - 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

AB - 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

KW - Fokker-planck equations

KW - Microscopic inclusions

KW - Navier-stokes equations

KW - Smoluchowski equations

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U2 - 10.4310/CMS.2005.v3.n4.a4

DO - 10.4310/CMS.2005.v3.n4.a4

M3 - Article

AN - SCOPUS:33745419628

SN - 1539-6746

VL - 3

SP - 531

EP - 544

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 4

ER -