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 & Continuous Dynamical Systems - A, 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). Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures,, Lectures in Mathematics, (2008). Google Scholar

[3]

D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials,, in Hyperbolic Problems-Theory, (2012), 136. Google Scholar

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615. Google Scholar

[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. doi: 10.4310/CMS.2010.v8.n1.a4. Google Scholar

[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. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[7]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbbR^n$,, Comm. Math. Phys., 274 (2007), 717. doi: 10.1007/s00220-007-0288-1. Google Scholar

[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). doi: 10.1142/S0218202511400057. Google Scholar

[9]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pure Appl. Math., 64 (2011), 45. doi: 10.1002/cpa.20334. Google Scholar

[10]

S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators,, Comm. Partial Diff. Eq., 36 (2011), 777. doi: 10.1080/03605302.2010.534224. Google Scholar

[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. doi: 10.1137/070683337. Google Scholar

[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. doi: 10.1051/cocv/2014032. Google Scholar

[13]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients,, Nonlinear Analysis TMA, 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1. Google Scholar

[14]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Eq., 24 (1999), 2173. doi: 10.1080/03605309908821498. Google Scholar

[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. Google Scholar

[16]

Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions,, Comm. Pure and Appl. Math., 44 (1991), 375. doi: 10.1002/cpa.3160440402. Google Scholar

[17]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Analysis RWA, 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002. Google Scholar

[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. doi: 10.1215/00127094-2010-211. Google Scholar

[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. doi: 10.4171/RMI/376. Google Scholar

[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). doi: 10.1142/S0218202511500230. Google Scholar

[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. doi: 10.1007/s00030-012-0164-3. Google Scholar

[22]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[23]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[24]

A. F. Filippov, Differential equations with discontinuous right-hand side,, A.M.S. Transl.(2), 42 (1964), 199. Google Scholar

[25]

F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws,, C. R. Acad. Sci. Paris, 349 (2011), 657. doi: 10.1016/j.crma.2011.05.004. Google Scholar

[26]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[27]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation,, SIAM J. Numer. Anal., 53 (2015), 895. doi: 10.1137/140959997. Google Scholar

[28]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[29]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[30]

H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows,, Arch. Rat. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. Google Scholar

[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. doi: 10.1007/s00285-004-0279-1. Google Scholar

[32]

J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations,, Arch. Rational Mech. Anal., 158 (2001), 29. doi: 10.1007/s002050100139. Google Scholar

[33]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Diff. Eq., 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[34]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[35]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar

[36]

F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients,, Comm. Partial Diff. Equ., 22 (1997), 337. doi: 10.1080/03605309708821265. Google Scholar

[37]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation,, Methods Appl. Anal., 9 (2002), 533. doi: 10.4310/MAA.2002.v9.n4.a4. Google Scholar

[38]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory,, Probab. Appl. (N. Y.), (1998). Google Scholar

[39]

C. Villani, Optimal Transport, Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[40]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003). Google Scholar

[41]

A. I. Vol'pert, The spaces BV and quasilinear equations,, Math. USSR Sb., 73 (1967), 255. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (2000). Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric space of probability measures,, Lectures in Mathematics, (2008). Google Scholar

[3]

D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials,, in Hyperbolic Problems-Theory, (2012), 136. Google Scholar

[4]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615. Google Scholar

[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. doi: 10.4310/CMS.2010.v8.n1.a4. Google Scholar

[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. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[7]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbbR^n$,, Comm. Math. Phys., 274 (2007), 717. doi: 10.1007/s00220-007-0288-1. Google Scholar

[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). doi: 10.1142/S0218202511400057. Google Scholar

[9]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pure Appl. Math., 64 (2011), 45. doi: 10.1002/cpa.20334. Google Scholar

[10]

S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators,, Comm. Partial Diff. Eq., 36 (2011), 777. doi: 10.1080/03605302.2010.534224. Google Scholar

[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. doi: 10.1137/070683337. Google Scholar

[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. doi: 10.1051/cocv/2014032. Google Scholar

[13]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients,, Nonlinear Analysis TMA, 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1. Google Scholar

[14]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Eq., 24 (1999), 2173. doi: 10.1080/03605309908821498. Google Scholar

[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. Google Scholar

[16]

Y. Brenier, Polar factorization and monotone rearrangement of vectorvalued functions,, Comm. Pure and Appl. Math., 44 (1991), 375. doi: 10.1002/cpa.3160440402. Google Scholar

[17]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Analysis RWA, 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002. Google Scholar

[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. doi: 10.1215/00127094-2010-211. Google Scholar

[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. doi: 10.4171/RMI/376. Google Scholar

[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). doi: 10.1142/S0218202511500230. Google Scholar

[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. doi: 10.1007/s00030-012-0164-3. Google Scholar

[22]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodyna.mic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[23]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[24]

A. F. Filippov, Differential equations with discontinuous right-hand side,, A.M.S. Transl.(2), 42 (1964), 199. Google Scholar

[25]

F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws,, C. R. Acad. Sci. Paris, 349 (2011), 657. doi: 10.1016/j.crma.2011.05.004. Google Scholar

[26]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[27]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equation,, SIAM J. Numer. Anal., 53 (2015), 895. doi: 10.1137/140959997. Google Scholar

[28]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[29]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[30]

H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows,, Arch. Rat. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. Google Scholar

[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. doi: 10.1007/s00285-004-0279-1. Google Scholar

[32]

J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations,, Arch. Rational Mech. Anal., 158 (2001), 29. doi: 10.1007/s002050100139. Google Scholar

[33]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Diff. Eq., 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[34]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[35]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar

[36]

F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients,, Comm. Partial Diff. Equ., 22 (1997), 337. doi: 10.1080/03605309708821265. Google Scholar

[37]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation,, Methods Appl. Anal., 9 (2002), 533. doi: 10.4310/MAA.2002.v9.n4.a4. Google Scholar

[38]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory,, Probab. Appl. (N. Y.), (1998). Google Scholar

[39]

C. Villani, Optimal Transport, Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[40]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003). Google Scholar

[41]

A. I. Vol'pert, The spaces BV and quasilinear equations,, Math. USSR Sb., 73 (1967), 255. Google Scholar

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