# American Institute of Mathematical Sciences

March  2011, 4(1): 227-258. doi: 10.3934/krm.2011.4.227

## Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system

 1 ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China 3 UPMC Université Paris 6, UMR 7598 LJLL, F-75005, Paris, France

Received  June 2010 Revised  December 2010 Published  January 2011

We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential.
Citation: José A. Carrillo, Renjun Duan, Ayman Moussa. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic & Related Models, 2011, 4 (1) : 227-258. doi: 10.3934/krm.2011.4.227
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