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

José A. Carrillo; Renjun Duan; Ayman Moussa

doi:10.3934/krm.2011.4.227

Article Contents

2011, Volume 4, Issue 1: 227-258. Doi: 10.3934/krm.2011.4.227

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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
2.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
3.
UPMC Université Paris 6, UMR 7598 LJLL, F-75005, Paris

Received: June 2010

Revised: December 2010

Published: March 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q84, 35Q31; Secondary: 35B35.

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