A Kinetic Flocking Model with Diffusion

Communications in Mathematical Physics

Volumn 300, Issue 1, 2010, Pages 95-145

  Duan, Renjun ^a   Fornasier, Massimo ^b   Toscani, Giuseppe ^c  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^c UNIVERSITY OF PAVIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77957161384     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-010-1110-z     Document Type: Article

References (28)

1. Arnold A., Markowich P., Toscani G., Unterreiter A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Part. Diff. Eqs. 26, 43-100 (2001).
2. Carrillo J. A., Fornasier M., Rosado J., Toscani G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42(1), 218-236 (2010).
3. Chuang Y.-L., D'Orsogna M. R., Marthaler D., Bertozzi A. L., Chayes L.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D 232, 33-47 (2007).
4. Cucker F., Smale S.: Emergent behavior in flocks. IEEE Transactions of Automatic Control 52, 852-862 (2007).
5. Degond P., Motsch S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Meth. Appl. Sci. 18(Suppl), 1193-1215 (2008).
6. Degond P., Motsch S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys. 131, 989-1021 (2008).
7. Desvillettes L., Villani C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54(1), 1-42 (2001).
8. Dolbeault J., Mouhot C., Schmeiser C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Acad. Sci. Paris 347, 511-516 (2009).
9. N). J. Diff. Eqs. 244, 3204-3234 (2007).
10. Duan R.-J.: Stability of the Boltzmann equation with potential forces on torus. Physica D: Nonlinear Phenomena 238, 1808-1820 (2009).
11. 3. Arch. Rat. Mech. Anal. (2010). doi: 10. 1007/s00205-010-0318-6.
12. Duan R.-J., Ukai S., Yang T.: A combination of energy method and spectral analysis for studies on systems for gas motions. Front. Math. China 4(2), 253-282 (2009).
13. Duan R.-J., Ukai S., Yang T., Zhao H.-J.: Optimal decay estimates on the linearized Boltzmann equation with time-dependent forces and their applications. Commun. Math. Phys. 277(1), 189-236 (2008).
14. Guo Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53, 1081-1094 (2004).
15. Guo Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391-434 (2002).
16. Guo Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104-1135 (2002).
17. Ha S.-Y., Liu J.-G.: A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7, 297-325 (2009).
18. Ha S.-Y., Tadmor E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic Rel. Models 1(3), 415-435 (2008).
19. Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto University, 1983.
20. Mouhot C., Neumann L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19, 969-998 (2006).
21. Topaz C., Bertozzi A. L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152-174 (2004).
22. Topaz C. M., Bertozzi A. L., Lewis M. A.: A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68, 1601-1623 (2006).
23. Ukai S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japa. Acad. 50, 179-184 (1974).
24. Ukai S., Yang T.: The Boltzmann equation in the space: Global and time-periodic solutions. Anal. Appl. 4, 263-310 (2006).
25. Vicsek T., Czirók A., Ben-Jacob E., Shochet O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226-1229 (1995).
26. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002, pp. 71-305.
27. Villani, C.: Hypocoercivity. Memoirs Amer. Math. Soc. 202, Providence, RI: Amer. Math. Soc. 2009, iv+141 pp.
28. Yates C., Erban R., Escudero C., Couzin L., Buhl J., Kevrekidis L., Maini P., Sumpter D.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Nat. Acad. Sci. 106(14), 5464-5469 (2009).