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chemotaxis; angiogenesis; degenerate parabolic equations; kinetic equations; global weak solutions; blow-up
Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia coli or amoeba like Dictyostelium discoïdeum exhibiting pointwise concentrations. For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted. The two systems can be unified by a kinetic approach that was proposed for Escherichia coli, based on more precise experiments showing a movement by ‘jump and tumble’. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells. We also mention the mathematical interest of analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels.
[1] W.  Alt: Biased random walk models for chemotaxis and related diffusion approximations. J.  Math. Biol. 9 (1980), 147–177. MR 0661424 | Zbl 0434.92001
[2] W.  Alt, G.  Hoffmann: Biological motion. Proceedings of a workshop held in Königswinter, Germany, March 16–19, 1989. Lecture Notes in Biomathematics, 89. Springer-Verlag, Berlin, 1990.
[3] A. R. A.  Anderson, M. A. J.  Chaplain: A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11 (1998), 109–114.
[4] C.  Bardos, R.  Santos, and R.  Sentis: Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284 (1984), 617–649. MR 0743736
[5] N.  Bellomo, L.  Preziosi: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modelling 32 (2000), 413–452. MR 1775113
[6] M. D.  Betterton, M. P.  Brenner: Collapsing bacterial cylinders. Phys. Rev.  E 64 (2001).
[7] P.  Biler: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9 (1999), 347–359. MR 1690388 | Zbl 0941.35009
[8] P.  Biler, T.  Nadzieja: A class of nonlocal parabolic problems occurring in statistical mechanics. Colloq. Math. 66 (1993), 131–145. MR 1242651
[9] P.  Biler, T.  Nadzieja: Global and exploding solutions in a model of self-gravitating systems. Rep. Math. Phys. 52 (2003), 205–225. MR 2016216
[10] M. P.  Brenner, P. Constantin, L. P.  Kadanoff, A. Schenkel, and S. C. Venkataramani: Diffusion, attraction and collapse. Nonlinearity 12 (1999), 1071–1098. MR 1709861
[11] M. P.  Brenner, L.  Levitov, and E. O.  Budrene: Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical Journal 74 (1995), 1677–1693.
[12] C.  Cercignani, R.  Illner, and M.  Pulvirenti: The Mathematical Theory of Dilute Gases. Applied Math. Sciences Vol. 106, Springer-Verlag, New York, 1994. MR 1307620
[13] F.  Chalub, P.  Markowich, B.  Perthame, and C.  Schmeiser: Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 142 (2004), 123–141. MR 2065025
[14] M. A. J.  Chaplain: Avascular growth, angiogenesis and vascular growth in solid tumors: the mathematical modelling of the stages of tumor development. Math. Comput. Modelling 23 (1996), 47–87.
[15] M. A. J.  Chaplain, L.  Preziosi: Macroscopic modelling of the growth and developement of tumor masses. Preprint No.  27, Politecnico di Torino, 2000.
[16] L. Corrias, B.  Perthame, and H.  Zaag: A chemotaxis model motivated by angiogenesis. C. R. Acad. Sci.  Paris, Ser.  I 336 (2003), 141–146. MR 1969568
[17] L. Corrias, B. Perthame, and H.  Zaag: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milano J.  Math. 72 (2004), 1–29. MR 2099126
[18] F. A.  Davidson, A. R. A.  Anderson, and M. A. J.  Chaplain: Steady-state solutions of a generic model for the formation of capillary networks. Appl. Math. Lett. 13 (2000), 127–132. MR 1760274
[19] P.  Degond, T.  Goudon, and F.  Poupaud: Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math.  J. 49 (2000), 1175–1198. MR 1803225
[20] Y.  Dolak, T.  Hillen: Cattaneo models for chemotaxis, numerical solution and pattern formation. J.  Math. Biol. 46 (2003), 461–478. MR 1963070
[21] J.  Dolbeault, B.  Perthame: Optimal critical mass in the two dimensional Keller-Segel model in $\mathbb{R}^2$. C. R. Acad. Sci. (2004) (to appear). MR 2103197
[22] Y.  Dolak, C.  Schmeiser: Kinetic Models for Chemotaxis. ANUM preprint. (2003). MR 2093271
[23] L. C.  Evans: Partial Differential Equations. Amer. Math. Soc., Providence, 1998. Zbl 0902.35002
[24] F.  Filbet, P.  Laurençot, and B.  Perthame: Derivation of hyperbolic models for chemosensitive movement. Preprint. Ecole Normale Supérieure, 2003. MR 2120548
[25] M. A.  Fontelos, A. Friedman, and B.  Hu: Mathematical analysis of a model for the initiation of angiogenesis. SIAM J.  Math. Anal. 33 (2002), 1330–1355. MR 1920634
[26] A.  Friedman, I.  Tello: Stability of solutions of chemotaxis equations in reinforced random walks. J.  Math. Anal. Appl. 272 (2002), 138–163. MR 1930708
[27] H.  Gajewski, K.  Zacharias: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195 (1998), 77–114. MR 1654677
[28] A.  Gamba, D.  Ambrosi, A.  Coniglio, A.  de Candia, S.  Di  Talia, E.  Giraudo, G.  Serini, L.  Preziosi, and F.  Bussolino: Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90 (2003), .
[29] I. Gasser, P.-E. Jabin, and B. Perthame: Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect.  A 130 (2000), 1259–1273. MR 1809103
[30] R. T.  Glassey: The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia, 1996. MR 1379589 | Zbl 0858.76001
[31] M. A.  Herrero, J. J. L.  Velázquez: Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996), 583–623. MR 1415081
[32] M. A.  Herrero, E. Medina, and J. J. L. Velázquez: Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10 (1997), 1739–1754. MR 1483563
[33] T.  Hillen, H.  Othmer: The diffusion limit of transport equations derived from velocity-jump processes. SIAM  J.  Appl. Math. 61 (2000), 751–775. MR 1788017
[34] D.  Horstmann: Lyapunov functions and $L^p$  estimates for a class of reaction-diffusion systems. Colloq. Math. 87 (2001), 113–127. MR 1812147
[35] D.  Horstmann: From  1970 until present: the Keller-Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math.-Ver. Vol. 105, 2003, pp. 103–165. MR 2013508 | Zbl 1071.35001
[36] H. J.  Hwang, K.  Kang, and A.  Stevens: Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. Math. Anal (to appear). MR 2139206
[37] W.  Jäger, S.  Luckhaus: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992), 819–824. MR 1046835
[38] E. F.  Keller: Assessing the Keller-Segel model: How has it fared? Biological growth and spread. Proc. Conf., Heidelberg,  1979. Lecture Notes in Biomath. Vol. 38, Springer-Verlag, Berlin-New York, 1980, pp. 379–387. MR 0609374
[39] E. F.  Keller, L. A.  Segel: Initiation of slime mold aggregation viewed as an instability. J.  Theoret. Biol. 26 (1970), 399–415.
[40] E. F.  Keller, L. A.  Segel: Model for chemotaxis. J.  Theoret. Biol. 30 (1971), 225–234.
[41] E. F.  Keller, L. A.  Segel: Travelling bands of chemotactic bacteria: a theoretical analysis. J.  Theoret. Biol. 30 (1971), 235–248.
[42] H. A.  Levine, B. D.  Sleeman: A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J.  Appl. Math. 57 (1997), 683–730. MR 1450846
[43] H. A.  Levine, B. D.  Sleeman: Partial differential equations of chemotaxis and angiogenesis. Math. Methods Appl. Sci. 24 (2001), 405–426. MR 1821934
[44] H. A.  Levine, M.  Nilsen-Hamilton, and B. D. Sleeman: Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42 (2001), 195–238. MR 1828815
[45] P. K.  Maini: Applications of mathematical modelling to biological pattern formation. Coherent Structures in Complex Systems (Sitges, 2000). Lecture Notes in Phys. Vol.  567, Springer-Verlag, Berlin, 2001, pp. 205–217. MR 1995108
[46] D.  Manoussaki: Modeling and simulation of the formation of vascular networks. ESAIM Proc. 12 (2002 (electronic)), 108–114.
[47] A.  Marrocco: 2D simulation of chemotactic bacteria agreggation. ESAIM: Math. Model. Numer. Anal. 37 (2003), 617–630. MR 2018433
[48] P.  Michel, S.  Mischler, and B.  Perthame: General entropy equations for structured population models and scattering. C.  R.  Acad. Sci. Paris (to appear). MR 2065377
[49] J. D.  Murray: Mathematical Biology, Vol.  2, third revised edition. Spatial Models and Biomedical Applications. Springer-Verlag, , 2003. MR 1952568
[50] T.  Nagai: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581–601. MR 1361006 | Zbl 0843.92007
[51] T.  Nagai, T.  Senba: Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis. Adv. Math. Sci. Appl. 8 (1998), 145–156. MR 1623326
[52] J. Nieto, F. Poupaud, and J.  Soler: High field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Rational. Mech. Anal. 158 (2001), 29–59. MR 1834113
[53] H. G.  Othmer, A. Stevens: Aggregation, blowup and collapse: the ABC’s of taxis in reinforced random walks. SIAM J.  Appl. Math. 57 (1997), 1044–1081. MR 1462051
[54] H. G. Othmer, S. R.  Dunbar, and W.  Alt: Models of dispersal in biological systems. J.  Math. Biol. 26 (1988), 263–298. MR 0949094
[55] C. S.  Patlak: Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311–338. MR 0081586
[56] B.  Perthame: Mathematical tools for kinetic equations. Bull.  Amer. Math. Soc. (NS) 41 (2004), 205–244. MR 2043752 | Zbl 1151.82351
[57] M.  Rascle: On a system of non-linear strongly coupled partial differential equations arising in biology. Proc. Conf. on  Ordinary and Partial Differential Equation. Lectures Notes in Math. Vol. 846, Everitt and Sleeman (eds.), Springer-Verlag, New-York, 1981, pp. 290–298.
[58] M.  Rascle, C.  Ziti: Finite time blow-up in some models of chemotaxis. J.  Math. Biol. 33 (1995), 388–414. MR 1320430
[59] G.  Serini, D.  Ambrosi, E.  Giraudo, A.  Gamba, L.  Preziosi, and F.  Bussolino: Modeling the early stages of vascular network assembly. The EMBO Journal 22 (2003), 1771–1779.
[60] T.  Sanba and T.  Suzuki: Weak solutions to a parabolic-elliptic system of chemotaxis. J.  Functional. Analysis 47 (2001), 17–51. MR 1909263
[61] H. R. Schwetlick: Travelling fronts for multidimensional nonlinear transport equations. Ann. Inst. H.  Poincaré. Anal. non Linéaire 17 (2000), 523–550. MR 1782743
[62] A.  Stevens: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J.  Appl. Math. 61 (2000), 183–212. MR 1776393 | Zbl 0963.60093
[63] A.  Stevens, M.  Schwelick: Work in preparation.
[64] M. I.  Weinstein: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983), 567–576. MR 0691044 | Zbl 0527.35023
[65] Y.  Yang, H.  Chen, and W.  Liu: On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis. SIAM J.  Math. Anal. 33 (2001), 763–785. MR 1884721
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