Critical Mass Phenomenon for a Chemotaxis Kinetic Model With Spherically Symmetric Initial Data

Bournaveas, Nikolaos; Calvez, Vincent

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 5, pp. 1871-1895.

[1] Blanchet A., Dolbeault J., Perthame B., Two-Dimensional Keller-Segel Model: Optimal Critical Mass and Qualitative Properties of the Solutions, Electron. J. Differential Equations 44 (2006), 32 pp. (electronic). | MR | Zbl

[2] Bobylev A. V., Carrillo J. A., Gamba I. M., On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Stat. Phys. 98 (2000) 743-773, J. Stat. Phys. 103 (2001) 1137-1138, erratum. | MR | Zbl

[3] Bolley F., Carrillo J. A., Tanaka Theorem for Inelastic Maxwell Models, Comm. Math. Phys. 276 (2007) 287-314. | MR | Zbl

[4] Bournaveas N., Calvez V., Gutiérrez S., Perthame B., Global Existence for a Kinetic Model of Chemotaxis Via Dispersion and Strichartz Estimates, Comm. Partial Differential Equations 33 (2008) 79-95. | MR | Zbl

[5] Bournaveas N., Calvez V., Global Existence for the Kinetic Chemotaxis Model Without Pointwise Memory Effects, and Including Internal Variables, Kinetic and Related Models 1 (2008) 29-48. | MR | Zbl

[6] Calvez V., Corrias L., The Parabolic-Parabolic Keller-Segel Model in $R^2$, Commun. Math. Sci. 6 (2008) 417-447. | MR | Zbl

[7] Calvez V., Perthame B., Sharifi Tabar M., Modified Keller-Segel System and Critical Mass for the Log Interaction Kernel, in: Stochastic Analysis and Partial Differential Equations, Contemp. Math., vol. 429, Amer. Math. Soc., Providence, RI, 2007, pp. 45-62. | MR | Zbl

[8] Castella F., Perthame B., Estimations De Strichartz Pour Les Équations De Transport Cinétique, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 535-540. | MR | Zbl

[9] Chalub F. A.C. C., Dolak-Struss Y., Markowich P. A., Oelz D., Schmeiser C., Soreff A., Model Hierarchies for Cell Aggregation by Chemotaxis, Math. Models Methods Appl. Sci. 16 (2006) 1173-1197. | MR | Zbl

[10] Chalub F. A.C. C., Markowich P. A., Perthame B., Schmeiser C., Kinetic Models for Chemotaxis and Their Drift-Diffusion Limits, Monatsh. Math. 142 (2004) 123-141. | MR | Zbl

[11] Chalub F. A.C. C., Rodrigues J. F., A Class of Kinetic Models for Chemotaxis With Threshold to Prevent Overcrowding, Port. Math. (N.S.) 63 (2006) 227-250. | MR | Zbl

[12] Chavanis P.-H., Sire C., Virial Theorem and Dynamical Evolution of Self-Gravitating Brownian Particles in an Unbounded Domain. I, II, Phys. Rev. E 73 (2006) 066103-066104.

[13] Corrias L., Perthame B., Zaag H., Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions, Milan J. Math. 72 (2004) 1-28. | MR | Zbl

[14] Erban R., Othmer H. G., From Individual to Collective Behavior in Bacterial Chemotaxis, SIAM J. Appl. Math. 65 (2004) 361-391. | MR | Zbl

[15] Erban R., Othmer H. G., Taxis Equations for Amoeboid Cells, J. Math. Biol. 54 (2007) 847-885. | MR | Zbl

[16] Escobedo M., Mischler S., On a Quantum Boltzmann Equation for a Gas of Photons, J. Math. Pures Appl. 80 (2001) 471-515. | MR | Zbl

[17] Gajewski H., Zacharias K., Global Behaviour of a Reaction-Diffusion System Modelling Chemotaxis, Math. Nachr. 195 (1998) 77-114. | MR | Zbl

[18] Glassey R. T., The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996. | MR | Zbl

[19] Glassey R. T., Schaeffer J., On Symmetric Solutions of the Relativistic Vlasov-Poisson System, Comm. Math. Phys. 101 (1985) 459-473. | MR | Zbl

[20] Hillen T., $M^5$ Mesoscopic and Macroscopic Models for Mesenchymal Motion, J. Math. Biol. 53 (2006) 585-616. | MR | Zbl

[21] Horst E., On the Classical Solutions of the Initial Value Problem for the Unmodified Nonlinear Vlasov Equation. II. Special Cases, Math. Methods Appl. Sci. 4 (1982) 19-32. | MR | Zbl

[22] Hwang H. J., Kang K., Stevens A., Global Solutions of Nonlinear Transport Equations for Chemosensitive Movement, SIAM J. Math. Anal. 36 (2005) 1177-1199. | MR | Zbl

[23] Jäger W., Luckhaus S., On Explosions of Solutions to a System of Partial Differential Equations Modelling Chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819-824. | MR | Zbl

[24] Lemou M., Méhats F., Raphael P., On the Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System, C.R. Math. Acad. Sci. Paris, Ser. I 341 (2005) 269-274. | MR | Zbl

[25] Lu X., The Boltzmann Equation for Bose-Einstein Particles: Velocity Concentration and Convergence to Equilibrium, J. Stat. Phys. 119 (2005) 1027-1067. | MR | Zbl

[26] Mischler S., Mouhot C., Rodriguez Ricard M., Cooling Process for Inelastic Boltzmann Equations for Hard Spheres. I. the Cauchy Problem, J. Stat. Phys. 124 (2006) 655-702. | MR | Zbl

[27] Nagai T., Blow-Up of Radially Symmetric Solutions to a Chemotaxis System, Adv. Math. Sci. Appl. 5 (1995) 581-601. | MR | Zbl

[28] Othmer H. G., Dunbar S. R., Alt W., Models of Dispersal in Biological Systems, J. Math. Biol. 26 (1988) 263-298. | MR | Zbl

[29] Othmer H. G., Hillen T., The Diffusion Limit of Transport Equations. II. Chemotaxis Equations, SIAM J. Appl. Math. 62 (2002) 1222-1250. | MR | Zbl

[30] Othmer H. G., Stevens A., Aggregation, Blowup, and Collapse: the ABCs of Taxis in Reinforced Random Walks, SIAM J. Appl. Math. 57 (1997) 1044-1081. | MR | Zbl

[31] Saint-Raymond L., Kinetic Models for Superfluids: a Review of Mathematical Results, C. R. Phys. 5 (2004) 65-75.