General nonlocal diffusion-convection mean field models: Nonexistence of global solutions

Physica A: Statistical Mechanics and its Applications

Volumn 379, Issue 2, 2007, Pages 523-533

  Biler, Piotr ^a   Woyczyński, Wojbor ^b  

^a UNIVERSITY OF WROC AW   (Poland)

^b CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

Collapse of solutions; Interacting particles; Mean field equation; Nonlocal diffusion convection; Self consistent field

Indexed keywords

DIFFUSION; FLOW INTERACTIONS; HEAT CONVECTION; MATHEMATICAL MODELS;

DIFFUSION OPERATORS; INTERACTING PARTICLES; MEAN FIELD EQUATION; SELF CONSISTENT FIELD;

MEAN FIELD THEORY;

EID: 34247507199     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2007.02.083     Document Type: Article

References (47)

1. Chavanis P.-H., Sommeria J., and Robert R. Statistical mechanics of two-dimensional vortices and collisionless stellar systems. Astrophys. J. 471 (1996) 385-399
2. Chavanis P.-H. Phase transitions in self-gravitating systems: self-gravitating fermions and hard sphere models. Phys. Rev. E 65 (2002) 056123
3. Chavanis P.-H. Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions. Phys. Rev. E 69 (2004) 066126
4. Biler P., and Stańczy R. Parabolic-elliptic systems with general density-pressure relations. In: Misawa M., and Suzuki T. (Eds). Proceedings of the Conference Variational Problems and Related Topics vol. 1405 (2004), Sūrikaisekikenkyūsho Kōkyūroku 31-53
5. Sugiyama Y. Global existence and finite time blow up for some degenerate Keller-Segel model. In: Naito Y., and Suzuki T. (Eds). Proceedings of the Conference Variational Problems and Related Topics, Kyoto 2005 vol. 1464 (2006), Sūrikaisekikenkyūsho Kōkyūroku
6. Biler P., and Dolbeault J. Long time behaviour of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems. Ann. Henri Poincaré 1 (2000) 461-472
7. Biler P., Dolbeault J., and Markowich P.A. Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling. Transport Theory Stat. Physics 30 (2001) 521-536
8. Jüngel A. Quasi-hydrodynamic semiconductor equations. PNLDE vol. 41 (2001), Birkhäuser, Basel, Boston
9. D. Horstmann, From until present: the Keller-Segel model in chemotaxis and its consequences I, II, Jahresber. Deutsch. Math.-Verein. 105 (2003) 103-165; 106 (2004) 51-69.
10. Córdoba A., and Córdoba D. A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249 (2004) 511-528
11. Biler P., and Woyczyński W.A. Global and exploding solutions of nonlocal quadratic evolution problems. SIAM J. Appl. Math. 59 (1999) 845-869
12. Biler P., Hebisch W., and Nadzieja T. The Debye system: existence and long time behavior of solutions. Nonlinear Anal. Theory Methods Appli. 23 (1994) 1189-1209
13. G. Karch, W.A. Woyczyński, Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc. (2007), to appear.
14. Escudero C. Chemotactic collapse and mesenchymal morphogenesis. Phys. Rev. E 72 (2005) 022903
15. C. Escudero, The fractional Keller-Segel model, 1-12, preprint arXiv: math.AP/0611496.
16. Biler P., Karch G., and Woyczyński W.A. Asymptotics for conservation laws involving Lévy diffusion generators. Stud. Math. 148 (2001) 171-192
17. Biler P., Karch G., and Woyczyński W.A. Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. Inst. H. Poincaré-Analyse Non Linéaire 18 (2001) 613-637
18. J. Rosiński, Tempering stable processes, Stochastic Processes Appl. (2007), to appear.
19. Streater R.F. Nonlinear heat equations. Rep. Math. Phys. 40 (1997) 557-564
20. Streater R.F. The Soret and Dufour effects in statistical dynamics. Proc. R. Soc. London A 456 (2000) 205-211
21. Biler P., Krzywicki A., and Nadzieja T. Self-interaction of Brownian particles coupled with thermodynamic processes. Rep. Math. Phys. 42 (1998) 359-372
22. Biler P., Nadzieja T., and Stańczy R. Nonisothermal systems of self-interacting Fermi-Dirac particles. Nonlocal Elliptic and Parabolic Problems, Proceedings of a Conference held in Bȩdlewo, 2003 vol. 66 (2004), Banach Center Publications, Polish Academy of Sciences, Warsaw 61-78
23. Stańczy R. Steady states for a system describing self-gravitating Fermi-Dirac particles. Differential Equations 18 (2005) 567-582
24. Raczyński A. Steady states for polytropic equation of self-gravitating gas. Math. Methods Appl. Sci. 28 (2005) 1881-1896
25. Wolansky G. Critical behaviour of semi-linear elliptic equations with sub-critical exponents. Nonlinear Anal. 26 (1996) 971-995
26. Biler P., Laurençot Ph., and Nadzieja T. On an evolution system describing self-gravitating Fermi-Dirac particles. Adv. Differential Equations 9 (2004) 563-586
27. Biler P., Dolbeault J., Esteban M.J., Markowich P.A., and Nadzieja T. Steady states for Streater's energy-transport models of self-gravitating particles. In: Ben Abdallah N., Arnold A., Degond P., et al. (Eds). Transport in Transition Regimes vol. 135 (2004), Springer IMA Volumes in Mathematics and its Applications 37-56
28. Biler P., and Nadzieja T. Global and exploding solutions in a model of self-gravitating systems. Rep. Math. Phys. 52 (2003) 205-225
29. Stańczy R. Self-attracting Fermi-Dirac particles in canonical and microcanonical setting. Math. Methods Appl. Sci. 28 (2005) 975-990
30. Biler P., Cannone M., Guerra I.A., and Karch G. Global regular and singular solutions for a model of gravitating particles. Math. Ann. 330 (2004) 693-708
31. P. Biler, G. Karch, Ph. Laurençot, T. Nadzieja, The 8 π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006) 1563-1583; published online: 27 Apr 2006; DOI: 10.1002/mma.743.
32. M. Olech, Ph.D. Thesis, University of Wrocław, in preparation.
33. Benachour S., Koch H., and Laurençot Ph. Very singular solutions to a nonlinear parabolic equation with absorption, II: Uniqueness. Proc. R. Soc. Edinburgh, Sect. A, Math. 134 (2004) 39-54
34. Brézis H., Peletier L.A., and Terman D. A very singular solution of the heat equation with absorption. Arch. Ration. Mech. Anal. 95 (1986) 185-209
35. Arnold A., Markowich P.A., Toscani G., and Unterreiter A. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Differential Equations 26 (2001) 43-100
36. Biler P., and Karch G. Generalized Fokker-Planck equations and convergence to their equilibria. Proceedings of the Conference Evolution Equations 2001 vol. 60 (2003), Banach Center Publications, Polish Academy of Sciences, Warsaw 307-318
37. Blanchet A., Dolbeault J., and Perthame B. Two dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differential Equations 44 (2006) 1-33
38. A.I. Saichev, W.A. Woyczyński, Model description of passive tracer density fields in the framework of Burgers' and other related model equations, in: T. Funaki et al. (Eds.), Nonlinear Stochastic PDEs. Hydrodynamic Limit and Burgers' Turbulence. Based on the Proceedings of a Workshop held at IMA, University of Minnesota, MN, USA during the 1993-94 IMA program on emerging applications of probability, Springer, Berlin, IMA Volumes in Mathematical Application, vol. 77 (1996) 167-191.
39. Biler P. Existence and nonexistence of solutions for a model of gravitational interaction of particles, III. Colloq. Math. 68 (1995) 229-239
40. Chavanis P.-H., and Sire C. Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain. Phys. Rev. E 73 (2006) 0660103-0660104
41. Herrero M.A., and Velázquez J.J.L. Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996) 583-623
42. Suzuki T. Free Energy and Self-Interacting Particles. PNLDE vol. 62 (2005), Birkhäuser, Boston
43. Herrero M.A., Medina E., and Velázquez J.J.L. Self-similar blow-up for a reaction-diffusion system. J. Comput. Appl. Math. 97 (1998) 99-119
44. Biler P., Karch G., Laurençot Ph., and Nadzieja T. The 8 π-problem for radially symmetric solutions of a chemotaxis model in a disc. Topol. Methods Nonlinear Anal. 27 (2006) 133-144
45. Stańczy R. On some parabolic-elliptic system with self-similar pressure term. "Self-Similar Solutions in Nonlinear PDE", Proceedings of a Conference held in Bȩdlewo, 2005 vol. 74 (2006), Banach Center Publications, Polish Academy of Sciences, Warsaw 205-215
46. Chavanis P.-H., and Sire C. Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions. Phys. Rev. E 69 (2004) 016116
47. Keller E.F., and Segel L.A. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biology 26 (1970) 399-415