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Singular Cucker–Smale Dynamics

  • Piotr Minakowski
  • Piotr B. Mucha
  • Jan PeszekEmail author
  • Ewelina Zatorska
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

This chapter is dedicated to the singular models of flocking. We give an overview of the existing literature starting from microscopic Cucker–Smale (CS) model with singular communication weight, through its mesoscopic mean-field limit, up to the corresponding macroscopic regime. For the microscopic CS model and its selected variants, the collision-avoidance phenomenon is discussed. For the kinetic mean-field model, we sketch the existence of global-in-time measure-valued solutions, paying special attention to weak-atomic uniqueness of solutions. Ultimately, for the macroscopic singular model, we provide a summary of existence results for the Euler-type alignment system. This includes the existence of strong solutions on a one-dimensional torus, and the extension of this result to higher dimensions by restricting the size of the initial data. Additionally, we present the pressureless Navier–Stokes-type system corresponding to particular choice of alignment kernel. This system is then compared—analytically and numerically—to the porous medium equation.

Notes

Acknowledgements

PM was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—314838170, GRK 2297 MathCoRe. JP was supported by the Polish MNiSW grant Mobilność Plus no. 1617/MOB/V/2017/0 and by the NSF grant RNMS11-07444 (KI-Net). EZ was supported by the UCL Department of Mathematics Grant, grant Iuventus Plus no. 0888/IP3/2016/74 of Ministry of Sciences and Higher Education RP, and by the Simons—Foundation grant 346300 and the Polish Government MNiSW 2015–2019 matching fund.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Piotr Minakowski
    • 1
  • Piotr B. Mucha
    • 2
  • Jan Peszek
    • 3
    • 4
    Email author
  • Ewelina Zatorska
    • 5
  1. 1.Institute of Analysis and NumericsOtto von Guericke University MagdeburgMagdeburgGermany
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland
  3. 3.Center for Scientific Computation and Mathematical Modeling (CSCAMM)University of MarylandCollege ParkUSA
  4. 4.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland
  5. 5.Department of MathematicsUniversity College LondonLondonUK

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