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Science China Mathematics

, Volume 61, Issue 11, pp 2033–2052 | Cite as

On the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system with shear thickening

  • Seung-Yeal Ha
  • Hwa Kil Kim
  • Jae-Myoung Kim
  • Jinyeong Park
Articles
  • 20 Downloads

Abstract

We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.

Keywords

emergence flocking Cucker-Smale model Navier-Stokes equation non-Newtonian fluid 

MSC(2010)

70F45 76A05 76D05 

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Notes

Acknowledgements

The work of Seung-Yeal Ha was supported by the Samsung Science and Technology Foundation (Grant No. SSTF-BA1401-03). The work of Hwa Kil Kim was supported by the National Research Foundation of Korea (Grant No. NRF2015R1D1A1A01056696). The work of Jae-Myoung Kim was supported by BK21 PLUS SNU Mathematical Sciences Division and the National Research Foundation of Korea (Grant No. NRF-2016R1D1A1B03930422).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Seung-Yeal Ha
    • 1
    • 2
  • Hwa Kil Kim
    • 3
  • Jae-Myoung Kim
    • 4
  • Jinyeong Park
    • 5
  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of Mathematics EducationHannam UniversityDaejeonRepublic of Korea
  4. 4.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea
  5. 5.Department of Mathematics and Research Institute of Natural SciencesHanyang UniversitySeoulRepublic of Korea

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