Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in ℝ2

Young Pil Choi; Jihoon Lee

doi:10.1016/j.nonrwa.2015.07.013

Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in ℝ²

Young Pil Choi, Jihoon Lee

Department of Mathematics

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

20 Citations (Scopus)

Abstract

Abstract We study the existence theory for the Cucker-Smale-Navier-Stokes (in short, CS-NS) equations in two dimensions. The CS-NS equations consist of Cucker-Smale flocking particles described by a Vlasov-type equation and incompressible Navier-Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS-NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.

Original language	English
Article number	2347
Pages (from-to)	158-182
Number of pages	25
Journal	Nonlinear Analysis: Real World Applications
Volume	27
DOIs	https://doi.org/10.1016/j.nonrwa.2015.07.013
Publication status	Published - 2016 Aug 17

Bibliographical note

Funding Information:
Y.-P. Choi was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2012R1A6A3A03039496 ). J. Lee was supported by an NRF grant funded by the Korea government (MEST) (No. 2009-0083521 ). The authors also want to thank professor José A. Carrillo and Seung-Yeal Ha for numerous helpful discussions.

Publisher Copyright:
© 2015 Elsevier Ltd.

All Science Journal Classification (ASJC) codes

Analysis
Engineering(all)
Economics, Econometrics and Finance(all)
Computational Mathematics
Applied Mathematics

Access to Document

10.1016/j.nonrwa.2015.07.013

Cite this

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title = "Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in ℝ2",

abstract = "Abstract We study the existence theory for the Cucker-Smale-Navier-Stokes (in short, CS-NS) equations in two dimensions. The CS-NS equations consist of Cucker-Smale flocking particles described by a Vlasov-type equation and incompressible Navier-Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS-NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.",

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AU - Choi, Young Pil

AU - Lee, Jihoon

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PY - 2016/8/17

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N2 - Abstract We study the existence theory for the Cucker-Smale-Navier-Stokes (in short, CS-NS) equations in two dimensions. The CS-NS equations consist of Cucker-Smale flocking particles described by a Vlasov-type equation and incompressible Navier-Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS-NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.

AB - Abstract We study the existence theory for the Cucker-Smale-Navier-Stokes (in short, CS-NS) equations in two dimensions. The CS-NS equations consist of Cucker-Smale flocking particles described by a Vlasov-type equation and incompressible Navier-Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS-NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.

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