Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force

Renjun Duan; Shuangqian Liu

doi:10.3934/krm.2013.6.687

Article Contents

2013, Volume 6, Issue 4: 687-700. Doi: 10.3934/krm.2013.6.687

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Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force

Renjun Duan^1, and
Shuangqian Liu^2,

1.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2.
Department of Mathematics, Jinan Unviersity, Guangdong

Received: February 28, 2013

Revised: May 31, 2013

Published: November 2013

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Abstract

We are concerned with a two-phase flow system consisting of the Vlasov-Fokker-Planck equation for particles coupled to the compressible Euler equations for the fluid through the friction force. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The proof is based on the classical energy estimates.

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

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