# American Institute of Mathematical Sciences

February  2010, 27(1): 237-263. doi: 10.3934/dcds.2010.27.237

## On the global smooth solution to 2-D fluid/particle system

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  July 2009 Revised  December 2009 Published  February 2010

In two space dimension, we show that the steady state the solution of fluid/particle system may tend to after a long time is completely determined by the initial total momentum. Based on this observation, we prove the global-in-time existence of the classical solutions for arbitrary initial data to the system that couples the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. By linearized method and Littlewood-Paley analysis, the exponential rate of the convergence toward steady state is obtained under some specific assumptions.
Citation: Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237
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