Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

Myeongju Chae; Kyungkeun Kang; Jihoon Lee

doi:10.1016/j.jde.2011.07.016

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

Myeongju Chae, Kyungkeun Kang, Jihoon Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

27 Citations (Scopus)

Abstract

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.

Original language	English
Pages (from-to)	2431-2465
Number of pages	35
Journal	Journal of Differential Equations
Volume	251
Issue number	9
DOIs	https://doi.org/10.1016/j.jde.2011.07.016
Publication status	Published - 2011 Nov 1

Bibliographical note

Funding Information:
The authors thank anonymous referee for his/her careful reading and valuable remarks. M. Chae’s work was supported by the National Research Foundation of Korea (NRF No. 2009-0069501). K. Kang’s work was partially supported by KRF-2008-331-C00024 and NRF No. 2009-00088692. J. Lee’s work was partially supported by the National Research Foundation of Korea (NRF No. 2009-0088692).

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/j.jde.2011.07.016

Cite this

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title = "Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations",

abstract = "We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.",

author = "Myeongju Chae and Kyungkeun Kang and Jihoon Lee",

note = "Funding Information: The authors thank anonymous referee for his/her careful reading and valuable remarks. M. Chae{\textquoteright}s work was supported by the National Research Foundation of Korea (NRF No. 2009-0069501). K. Kang{\textquoteright}s work was partially supported by KRF-2008-331-C00024 and NRF No. 2009-00088692. J. Lee{\textquoteright}s work was partially supported by the National Research Foundation of Korea (NRF No. 2009-0088692).",

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AU - Kang, Kyungkeun

AU - Lee, Jihoon

N1 - Funding Information: The authors thank anonymous referee for his/her careful reading and valuable remarks. M. Chae’s work was supported by the National Research Foundation of Korea (NRF No. 2009-0069501). K. Kang’s work was partially supported by KRF-2008-331-C00024 and NRF No. 2009-00088692. J. Lee’s work was partially supported by the National Research Foundation of Korea (NRF No. 2009-0088692).

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N2 - We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.

AB - We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.

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Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

Abstract

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