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

Uniqueness for Keller-Segel-type chemotaxis models

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

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.
Citation: José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319
References:
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P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.

[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.

[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.

[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.

[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 (). 

[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.

[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.

[15]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. Math. Sci., 6 (2008), 417-447.

[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.

[17]

J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, preprint, (). 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9), 2 (2009), 509-528.

[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.

[31]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,'' Cambridge Texts Appl. Math., Vol. 27, Cambridge Univ. Press, Cambridge, 2002.

[32]

R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[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.

[34]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

[35]

V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zhurn. Vych. Mat., 3 (1963), 1032-1066.

[36]

A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76. doi: 10.1215/S0012-7094-45-01206-3.

[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.

show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.

[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.

[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.

[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.

[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 (). 

[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.

[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.

[15]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. Math. Sci., 6 (2008), 417-447.

[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.

[17]

J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, preprint, (). 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9), 2 (2009), 509-528.

[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.

[31]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,'' Cambridge Texts Appl. Math., Vol. 27, Cambridge Univ. Press, Cambridge, 2002.

[32]

R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[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.

[34]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

[35]

V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zhurn. Vych. Mat., 3 (1963), 1032-1066.

[36]

A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76. doi: 10.1215/S0012-7094-45-01206-3.

[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.

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