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Optimal location problems with routing cost
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 |
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. |
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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