Uniqueness for Keller-Segel-type chemotaxis models

José Antonio Carrillo; Stefano Lisini; Edoardo Mainini

doi:10.3934/dcds.2014.34.1319

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April 2014, 34(4): 1319-1338. doi: 10.3934/dcds.2014.34.1319

Uniqueness for Keller-Segel-type chemotaxis models

José Antonio Carrillo ^1, , Stefano Lisini ^2, and Edoardo Mainini ^3,

1.	Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
2.	Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
3.	Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti (DIME), Università degli Studi di Genova, P.le Kennedy 1, 16129 Genova, Italy

Received December 2012 Revised April 2013 Published October 2013

Abstract
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We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

Keywords: Chemotaxis, displacement convexity., Wasserstein distance, Gradient flows, Keller-Segel model.

Mathematics Subject Classification: 35A02, 35K45, 35Q9.

Citation: José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319

References:

[1]	P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457. Google Scholar
[2]	L. Ambrosio and N. Gigli, A user's guide to optimal transport, in "Modelling and Optimisation of Flows on Networks,'' Lecture Notes in Mathematics, Springer, Berlin-Heidelberg, (2013), 1-155. doi: 10.1007/978-3-642-32160-3_1. Google Scholar
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[5]	L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223. Google Scholar
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[8]	P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. Google Scholar
[9]	A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar
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[13]	A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $R^d$, $d\ge 3$, Communication in Partial Differential Equations, 38 (2013), 658-686. doi: 10.1080/03605302.2012.757705. Google Scholar
[14]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. Google Scholar
[15]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. Math. Sci., 6 (2008), 417-447. Google Scholar
[16]	S. Campanato, Equazioni paraboliche del secondo ordine e spazi $L^{2,\theta}(\Omega,\delta)$, (Italian), Ann. Mat. Pura Appl. (4), 73 (1966), 55-102. doi: 10.1007/BF02415082. Google Scholar
[17]	J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, preprint, (). Google Scholar
[18]	J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the $2$-Wasserstein length space and thermalization of granular media, Arch. Rat. Mech. Anal., 179 (2006), 217-263. doi: 10.1007/s00205-005-0386-1. Google Scholar
[19]	J. A. Carrillo and J. Rosado, Uniqueness of bounded solutions to aggregation equations by optimal transport methods, in "European Congress of Mathematics,'' Eur. Math. Soc., Zürich, (2010), 3-16. doi: 10.4171/077-1/1. Google Scholar
[20]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar
[21]	L. Corrias, B. Perthame and H. Zaag, $L^p$ and $L^\infty$ a priori estimates for some chemotaxis models and applications to the Cauchy problem, in "The Mechanism of the Spatio-Temporal Pattern Arising in Reaction Diffusion System,'' Kyoto, 2004. Google Scholar
[22]	S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X. Google Scholar
[23]	I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584. Google Scholar
[24]	A. N. Konënkov, The Cauchy problem for the heat equation in Zygmund spaces, (Russian) Differ. Uravn., 41 (2005), 820-831, 863; translation in Differ. Equ., 41 (2005), 860-872. doi: 10.1007/s10625-005-0225-z. Google Scholar
[25]	R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar
[26]	R. Kowalczyk and Z. Szymańska}, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar
[27]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'' (Russian), Nauka, Moscow, 1968; Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. Google Scholar
[28]	G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded densitiy, J. Math. Pures Appl. (9), 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005. Google Scholar
[29]	E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9), 2 (2009), 509-528. Google Scholar
[30]	E. Mainini, Well-posedness for a mean field model of Ginzburg-Landau vortices with opposite degrees, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 133-158. doi: 10.1007/s00030-011-0121-6. Google Scholar
[31]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,'' Cambridge Texts Appl. Math., Vol. 27, Cambridge Univ. Press, Cambridge, 2002. Google Scholar
[32]	R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar
[33]	S. Serfaty and J. L. Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators,, to appear in Calc. Var. PDEs., (). doi: 10.1007/s00526-013-0613-9. Google Scholar
[34]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. Google Scholar
[35]	V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zhurn. Vych. Mat., 3 (1963), 1032-1066. Google Scholar
[36]	A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76. doi: 10.1215/S0012-7094-45-01206-3. Google Scholar
[37]	A. Zygmund, "Trigonometric Series,'' Vol. I, II, Third edition, With a foreword by Robert A. Fefferman, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. Google Scholar

show all references

References:

[1]	P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457. Google Scholar
[2]	L. Ambrosio and N. Gigli, A user's guide to optimal transport, in "Modelling and Optimisation of Flows on Networks,'' Lecture Notes in Mathematics, Springer, Berlin-Heidelberg, (2013), 1-155. doi: 10.1007/978-3-642-32160-3_1. Google Scholar
[3]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Spaces of Probability Measures,'' Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. Google Scholar
[4]	L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217-246. doi: 10.1016/j.anihpc.2010.11.006. Google Scholar
[5]	L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223. Google Scholar
[6]	A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Mod. Meth. Appl. Sci., 22 (2012), 1140005, 39 pp. doi: 10.1142/S0218202511400057. Google Scholar
[7]	A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334. Google Scholar
[8]	P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. Google Scholar
[9]	A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar
[10]	A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Func. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012. Google Scholar
[11]	A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. Google Scholar
[12]	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, 2006 (). Google Scholar
[13]	A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $R^d$, $d\ge 3$, Communication in Partial Differential Equations, 38 (2013), 658-686. doi: 10.1080/03605302.2012.757705. Google Scholar
[14]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. Google Scholar
[15]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. Math. Sci., 6 (2008), 417-447. Google Scholar
[16]	S. Campanato, Equazioni paraboliche del secondo ordine e spazi $L^{2,\theta}(\Omega,\delta)$, (Italian), Ann. Mat. Pura Appl. (4), 73 (1966), 55-102. doi: 10.1007/BF02415082. Google Scholar
[17]	J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, preprint, (). Google Scholar
[18]	J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the $2$-Wasserstein length space and thermalization of granular media, Arch. Rat. Mech. Anal., 179 (2006), 217-263. doi: 10.1007/s00205-005-0386-1. Google Scholar
[19]	J. A. Carrillo and J. Rosado, Uniqueness of bounded solutions to aggregation equations by optimal transport methods, in "European Congress of Mathematics,'' Eur. Math. Soc., Zürich, (2010), 3-16. doi: 10.4171/077-1/1. Google Scholar
[20]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar
[21]	L. Corrias, B. Perthame and H. Zaag, $L^p$ and $L^\infty$ a priori estimates for some chemotaxis models and applications to the Cauchy problem, in "The Mechanism of the Spatio-Temporal Pattern Arising in Reaction Diffusion System,'' Kyoto, 2004. Google Scholar
[22]	S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X. Google Scholar
[23]	I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584. Google Scholar
[24]	A. N. Konënkov, The Cauchy problem for the heat equation in Zygmund spaces, (Russian) Differ. Uravn., 41 (2005), 820-831, 863; translation in Differ. Equ., 41 (2005), 860-872. doi: 10.1007/s10625-005-0225-z. Google Scholar
[25]	R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar
[26]	R. Kowalczyk and Z. Szymańska}, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar
[27]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'' (Russian), Nauka, Moscow, 1968; Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1967. Google Scholar
[28]	G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded densitiy, J. Math. Pures Appl. (9), 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005. Google Scholar
[29]	E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9), 2 (2009), 509-528. Google Scholar
[30]	E. Mainini, Well-posedness for a mean field model of Ginzburg-Landau vortices with opposite degrees, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 133-158. doi: 10.1007/s00030-011-0121-6. Google Scholar
[31]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,'' Cambridge Texts Appl. Math., Vol. 27, Cambridge Univ. Press, Cambridge, 2002. Google Scholar
[32]	R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar
[33]	S. Serfaty and J. L. Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators,, to appear in Calc. Var. PDEs., (). doi: 10.1007/s00526-013-0613-9. Google Scholar
[34]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. Google Scholar
[35]	V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zhurn. Vych. Mat., 3 (1963), 1032-1066. Google Scholar
[36]	A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76. doi: 10.1215/S0012-7094-45-01206-3. Google Scholar
[37]	A. Zygmund, "Trigonometric Series,'' Vol. I, II, Third edition, With a foreword by Robert A. Fefferman, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. Google Scholar

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