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Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
1. | CNRS UMR 5669 "Unité de Mathématiques Pures et Appliquées", and project-team Inria NUMED, Ecole Normale Supérieure de Lyon, Lyon |
2. | Project-team MEPHYSTO, Inria Lille - Nord Europe, Villeneuve d'Ascq, France |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space Of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher, Séminaire Équations aux Dérivées Partielles, (2011-2012), 26pp.
doi: 10.5802/slsedp.6. |
[3] |
A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.
doi: 10.1137/070683337. |
[4] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.
doi: 10.1002/cpa.20225. |
[5] |
A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32pp (electronic). |
[6] |
V. Calvez and J. A. Carrillo, Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities, Proc. Amer. Math. Soc., 140 (2012), 3515-3530.
doi: 10.1090/S0002-9939-2012-11306-1. |
[7] |
V. Calvez, B. Perthame and M. Sharifi tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, in Stochastic Analysis and Partial Differential Equations, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62.
doi: 10.1090/conm/429/08229. |
[8] |
J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009/10), 4305-4329.
doi: 10.1137/080739574. |
[9] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[10] |
A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie, Ph.D thesis, Université Lille 1, 2010. |
[11] |
J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121.
doi: 10.3934/dcds.2009.25.109. |
[12] |
F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
doi: 10.1007/s00211-006-0024-3. |
[13] |
T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 2012. |
[14] |
Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[15] |
L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227 (electronic).
doi: 10.1137/050628015. |
[16] |
J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.
doi: 10.1007/s10955-009-9717-1. |
[17] |
J. Haškovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system, Comm. Partial Differential Equations, 36 (2011), 940-960.
doi: 10.1080/03605302.2010.538783. |
[18] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683 (1998). |
[19] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[20] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[21] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[22] |
N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008/09), 1852-1881.
doi: 10.1137/080722229. |
[23] |
E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[24] |
E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[25] |
S. Luckhaus, Y. Sugiyama and J. J. L. Velázquez, Measure valued solutions of the 2D Keller-Segel system, Arch. Ration. Mech. Anal., 206 (2012), 31-80.
doi: 10.1007/s00205-012-0549-9. |
[26] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u$, Duke Math. J., 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[27] |
F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450.
doi: 10.3934/dcds.2002.8.435. |
[28] |
N. Mittal, E. O. Budrene, M. P. Brenner and A. V. Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proc. Natl. Acad. Sci. USA, 100 (2003), 13259-13263.
doi: 10.1073/pnas.2233626100. |
[29] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[30] |
P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation, arXiv:1209.2517. |
[31] |
T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50. |
[32] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[33] |
J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., 62 (2002), 1581-1633 (electronic).
doi: 10.1137/S0036139900380049. |
[34] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223 (electronic).
doi: 10.1137/S0036139903433888. |
[35] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248 (electronic).
doi: 10.1137/S003613990343389X. |
[36] |
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space Of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher, Séminaire Équations aux Dérivées Partielles, (2011-2012), 26pp.
doi: 10.5802/slsedp.6. |
[3] |
A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.
doi: 10.1137/070683337. |
[4] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.
doi: 10.1002/cpa.20225. |
[5] |
A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32pp (electronic). |
[6] |
V. Calvez and J. A. Carrillo, Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities, Proc. Amer. Math. Soc., 140 (2012), 3515-3530.
doi: 10.1090/S0002-9939-2012-11306-1. |
[7] |
V. Calvez, B. Perthame and M. Sharifi tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, in Stochastic Analysis and Partial Differential Equations, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62.
doi: 10.1090/conm/429/08229. |
[8] |
J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009/10), 4305-4329.
doi: 10.1137/080739574. |
[9] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[10] |
A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie, Ph.D thesis, Université Lille 1, 2010. |
[11] |
J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121.
doi: 10.3934/dcds.2009.25.109. |
[12] |
F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
doi: 10.1007/s00211-006-0024-3. |
[13] |
T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 2012. |
[14] |
Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[15] |
L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227 (electronic).
doi: 10.1137/050628015. |
[16] |
J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.
doi: 10.1007/s10955-009-9717-1. |
[17] |
J. Haškovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system, Comm. Partial Differential Equations, 36 (2011), 940-960.
doi: 10.1080/03605302.2010.538783. |
[18] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683 (1998). |
[19] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[20] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[21] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[22] |
N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008/09), 1852-1881.
doi: 10.1137/080722229. |
[23] |
E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[24] |
E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[25] |
S. Luckhaus, Y. Sugiyama and J. J. L. Velázquez, Measure valued solutions of the 2D Keller-Segel system, Arch. Ration. Mech. Anal., 206 (2012), 31-80.
doi: 10.1007/s00205-012-0549-9. |
[26] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u$, Duke Math. J., 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[27] |
F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450.
doi: 10.3934/dcds.2002.8.435. |
[28] |
N. Mittal, E. O. Budrene, M. P. Brenner and A. V. Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proc. Natl. Acad. Sci. USA, 100 (2003), 13259-13263.
doi: 10.1073/pnas.2233626100. |
[29] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[30] |
P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation, arXiv:1209.2517. |
[31] |
T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50. |
[32] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[33] |
J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., 62 (2002), 1581-1633 (electronic).
doi: 10.1137/S0036139900380049. |
[34] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223 (electronic).
doi: 10.1137/S0036139903433888. |
[35] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248 (electronic).
doi: 10.1137/S003613990343389X. |
[36] |
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
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