Critical thresholds in flocking hydrodynamics with non-local alignment

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 372, Issue 2028, 2014, Pages

  Tadmor, Eitan ^a,b   Tan, Changhui ^a,b  

^a UNIVERSITY OF MARYLAND   (United States)

^b Bldg 142   (United States)

Author keywords

Alignment; Critical thresholds; Flocking; Hydrodynamics; Regularity

Indexed keywords

COMPUTATIONAL METHODS; HYDRODYNAMICS; PHASE SPACE METHODS;

AGENT-BASED MODEL; CRITICAL THRESHOLD; FINITE TIME BLOW-UP; FLOCKING; GLOBAL REGULARITIES; HYDRODYNAMIC DESCRIPTION; INITIAL CONFIGURATION; REGULARITY;

ALIGNMENT;

EID: 84907870339     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2013.0401     Document Type: Article

References (22)

1. Ha S-Y, Tadmor E. 2008 From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic Related Models 1, 415-435. (doi:10.3934/krm.2008.1.415)
2. Vicsek T, Zefeiris A. 2012 Collective motion. Phys. Reprint. 517, 71-140. (doi:10.1016/j.physrep.2012.03.004)
3. Motsch S, Tadmor E. In press. Heterophilious dynamics enhances consensus. SIAM Rev.
4. Cucker F, Smale S. 2007 Emergent behavior in flocks. IEEE Trans. Autom. Control 52, 852-862. (doi:10.1109/TAC.2007.895842)
5. Cucker F, Smale S. 2007 On the mathematics of emergence. Jpn J. Math. 2, 197-227. (doi:10.1007/s11537-007-0647-x)
6. Motsch S, Tadmor E. 2011 A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144, 923-947. (doi:10.1007/s10955-011-0285-9)
7. Liu T-P, Xin Z, Yang T. 1998 Vacuum states of compressible flow. Discrete Contin. Dyn. Syst. 4, 1-32.
8. Yang T, Yao Z, Zhu C. 2001 Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Commun. Partial Diff. Equ. 26, 965-981. (doi:10.1081/PDE-100002385)
9. Yang T, Zhu C. 2002 Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329-363. (doi:10.1007/s00220-002-0703-6)
10. Mellet A, Vasseur A. 2007/08 Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal. 39, 1344-1365. (doi:10.1137/060658199)
11. Katz N, Pavlovic N. 2002 A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation. Geom. Funct. Anal. 12, 355-379. (doi:10.1007/s00039-002-8250-z)
12. Wu J. 2003 Generalized MHD equations. J. Diff. Equ. 195, 284-312. (doi:10.1016/j.jde.2003.07.007)
13. Caffarelli L, Vasseur A. 2010 Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903-1930. (doi:10.4007/annals.2010.171.1903)
14. Kiselev A, Nazarov F, Sterenberg R. 2008 Blow up and regularity for fractal Burgers equation. Dyn. Partial Diff. Equ. 5, 211-240. (doi:10.4310/DPDE.2008.v5.n3.a2)
15. Constantin P, Vicol V. 2012 Nonlinear maximum principles for dissipative linear nonlocal operators and applications. GAFA 22, 1289-1321.
16. Ha S-Y, Liu J-G. 2009 A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7, 297-325. (doi:10.4310/CMS.2009.v7.n2.a2)
17. Engelberg S, Liu H, Tadmor E. 2001 Critical threshold in Euler-Poisson equations. Indiana Univ. Math. J. 50, 109-157. (doi:10.1512/iumj.2001.50.2177)
18. Liu H, Tadmor E. 2002 Spectral dynamics of velocity gradient field in restricted flows. Commun. Math. Phys. 228, 435-466. (doi:10.1007/s002200200667)
19. Liu H, Tadmor E. 2003 Critical threshold in 2D restricted Euler-Poisson equations. SIAM J. Math. Anal. 63, 1889-1910. (doi:10.1137/S0036139902416986)
20. Cheng B, Tadmor E. 2009 An improved local blow-up condition for Euler-Poisson equations with attractive forcing. Physica D 238, 2062-2066. (doi:10.1016/j.physd.2009.08.008)
21. Tadmor E, Wei D. 2008 On the global regularity of sub-critical Euler-Poisson equations with pressure. J. Eur. Math. Soc. 10, 757-769. (doi:10.4171/JEMS/129)
22. Liu H, Tadmor E. 2001 Critical thresholds in convolution model for nonlinear conservation laws. SIAM J. Math. Anal. 33, 930-945. (doi:10.1137/S0036141001386908)