# American Institute of Mathematical Sciences

March  2016, 36(3): 1355-1382. doi: 10.3934/dcds.2016.36.1355

## Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

 1 Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans & CNRS UMR 7349, Fédération Denis Poisson, Université d'Orléans & CNRS FR 2964, 45067 Orléans Cedex 2, France 2 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, INRIA Paris-Rocquencourt, EPI MAMBA, F-75005, Paris

Received  January 2014 Revised  June 2015 Published  August 2015

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a self-consistent interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)) [18], a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developed. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developed in the spirit of James and Vauchelet (NoDEA (2013)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
Citation: François James, Nicolas Vauchelet. Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1355-1382. doi: 10.3934/dcds.2016.36.1355
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures, Lectures in Mathematics, Birkäuser, 2008. [3] D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials, in Hyperbolic Problems-Theory, Numerics and Applications, Vol. 1, Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 136-147. [4] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. [5] A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4. [6] A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [7] A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbb{R}^{N}$, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1. [8] A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp. doi: 10.1142/S0218202511400057. [9] 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. [10] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796. doi: 10.1080/03605302.2010.534224. [11] 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. [12] G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441. doi: 10.1051/cocv/2014032. [13] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933. doi: 10.1016/S0362-546X(97)00536-1. [14] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [15] F. Bouchut, F. James and S. Mancini, Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 4 (2005), 1-25. [16] Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions, Comm. Pure and Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [17] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002. [18] J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211. [19] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Matematica Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. [20] R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230. [21] G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523-537. doi: 10.1007/s00030-012-0164-3. [22] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6. [23] F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [24] A. F. Filippov, Differential equations with discontinuous right-hand side, A.M.S. Transl.(2), 42 (1964), 199-231. [25] F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661. doi: 10.1016/j.crma.2011.05.004. [26] F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4. [27] F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation, SIAM J. Numer. Anal., 53 (2015), 895-916. doi: 10.1137/140959997. [28] 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. [29] E. F. Keller and L. A. 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. [30] H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8. [31] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1. [32] J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [33] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Diff. Eq., 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [34] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6. [35] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [36] F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358. doi: 10.1080/03605309708821265. [37] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. doi: 10.4310/MAA.2002.v9.n4.a4. [38] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998. [39] C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009. doi: 10.1007/978-3-540-71050-9. [40] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc, Providence, 2003. [41] A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sb., 73 (1967), 255-302.

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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures, Lectures in Mathematics, Birkäuser, 2008. [3] D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials, in Hyperbolic Problems-Theory, Numerics and Applications, Vol. 1, Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 136-147. [4] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. [5] A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4. [6] A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [7] A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbb{R}^{N}$, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1. [8] A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp. doi: 10.1142/S0218202511400057. [9] 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. [10] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796. doi: 10.1080/03605302.2010.534224. [11] 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. [12] G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441. doi: 10.1051/cocv/2014032. [13] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933. doi: 10.1016/S0362-546X(97)00536-1. [14] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [15] F. Bouchut, F. James and S. Mancini, Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 4 (2005), 1-25. [16] Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions, Comm. Pure and Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [17] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002. [18] J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211. [19] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Matematica Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. [20] R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230. [21] G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523-537. doi: 10.1007/s00030-012-0164-3. [22] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6. [23] F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [24] A. F. Filippov, Differential equations with discontinuous right-hand side, A.M.S. Transl.(2), 42 (1964), 199-231. [25] F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661. doi: 10.1016/j.crma.2011.05.004. [26] F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4. [27] F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation, SIAM J. Numer. Anal., 53 (2015), 895-916. doi: 10.1137/140959997. [28] 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. [29] E. F. Keller and L. A. 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. [30] H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8. [31] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1. [32] J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [33] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Diff. Eq., 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [34] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6. [35] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [36] F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358. doi: 10.1080/03605309708821265. [37] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. doi: 10.4310/MAA.2002.v9.n4.a4. [38] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998. [39] C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009. doi: 10.1007/978-3-540-71050-9. [40] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, Amer. Math. Soc, Providence, 2003. [41] A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sb., 73 (1967), 255-302.
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