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On Strong Local Alignment in the Kinetic Cucker-Smale Model

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Hyperbolic Conservation Laws and Related Analysis with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 49))

Abstract

In this paper, we rigorously derive a kinetic Cucker-Smale model with strong local alignment. The local alignment term is obtained by considering the limit of a non-local alignment term recently proposed by Motsch and Tadmor. The main difficulty in the analysis is presented by the non-symmetry of the Motsch-Tadmor term as well as the behavior of the velocity when the density vanishes (vacuum). Tools involved are the averaging lemma and several L p estimates.

1991 Mathematics Subject Classification Primary: 35Q84; Secondary: 35D30

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References

  1. J.A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Cucker, S. Smale, Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)

    Article  MathSciNet  Google Scholar 

  3. F. Cucker, S. Smale, On the mathematics of emergence. Jpn. J. Math. 2(1), 197–227 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.-Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(30), 415–435 (2008)

    MathSciNet  MATH  Google Scholar 

  5. T. Karper, A. Mellet, K. Trivisa, Existence of weak solutions to kinetic flocking models. SIAM J. Math. Anal. 45(1), 215–243 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Karper, A. Mellet, K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking with strong local alignment (2012, Preprint)

    Google Scholar 

  7. S. Motsch, E. Tadmor, A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 141(5), 923–947 (2011)

    Article  MathSciNet  Google Scholar 

  8. B. Perthame, P.E. Souganidis, A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. 31(4), 591–598 (1998)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of Trygve K. Karper was supported by the Research Council of Norway through the project 205738. The work of Antoine Mellet was supported by the National Science Foundation under the Grant DMS-0901340. The work of Konstantina Trivisa was supported by the National Science Foundation under the Grant DMS-1109397.

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Authors and Affiliations

  1. Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD, 20742, USA

    Trygve K. Karper

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Antoine Mellet & Konstantina Trivisa

Authors
  1. Trygve K. Karper
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  2. Antoine Mellet
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  3. Konstantina Trivisa
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Correspondence to Trygve K. Karper .

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Editors and Affiliations

  1. University of Oxford Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  3. University of Oslo Centre of Mathematics for Applications, Oslo, Norway

    Kenneth H. Karlsen

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Karper, T.K., Mellet, A., Trivisa, K. (2014). On Strong Local Alignment in the Kinetic Cucker-Smale Model. In: Chen, GQ., Holden, H., Karlsen, K. (eds) Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics & Statistics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39007-4_11

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