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 -