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.
Bibliographical noteFunding 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.
© 2015 Elsevier Ltd.
All Science Journal Classification (ASJC) codes
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics