Communications in Mathematical Physics

, Volume 270, Issue 3, pp 789-811

First online:

Regularity of Coupled Two-Dimensional Nonlinear Fokker-Planck and Navier-Stokes Systems

  • P. ConstantinAffiliated withDepartment of Mathematics, The University of Chicago Email author 
  • , C. FeffermanAffiliated withDepartment of Mathematics, Princeton University
  • , E. S. TitiAffiliated withDepartment of Mathematics and Department of Mechanics and Aerospace Engineering, University of CaliforniaDepartment of Computer Science and Applied Mathematics, Weizmann Institute of Science
  • , A. ZarnescuAffiliated withDepartment of Mathematics, The University of Chicago

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We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the particles. Because the particles perform rapid random motion, we assume that the density of particles is carried by a time average of the fluid velocity. The resulting coupled system is shown to have smooth solutions at all values of parameters, in two spatial dimensions.