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Global Classical Solutions Close to Equilibrium to the Vlasov Euler-Fokker-Planck System

Ayman Moussa 1, 2 José Antonio Carrillo 3 Renjun Duan 4
2 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
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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Submitted on : Thursday, December 15, 2011 - 12:38:54 PM
Last modification on : Friday, March 27, 2020 - 4:01:45 AM

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Ayman Moussa, José Antonio Carrillo, Renjun Duan. Global Classical Solutions Close to Equilibrium to the Vlasov Euler-Fokker-Planck System. Kinetic and Related Models , AIMS, 2011, 4 (1), pp.227-258. ⟨10.3934/krm.2011.4.227⟩. ⟨hal-00652339⟩



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