February  2020, 40(2): 1191-1231. doi: 10.3934/dcds.2020075

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

2. 

DISIM - Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio 1 (Coppito) 67100 L'Aquila (AQ), Italy

* Corresponding author: José A. Carrillo

Received  June 2019 Revised  September 2019 Published  November 2019

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space $ L^2(0,1)^2 $ according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the $ L^m $-norms for all $ m\in [1,+\infty] $ and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

Citation: José Antonio Carrillo, Marco Di Francesco, Antonio Esposito, Simone Fagioli, Markus Schmidtchen. Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1191-1231. doi: 10.3934/dcds.2020075
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

[2]

L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 1-136.  doi: 10.1016/S1874-5717(07)80004-1.

[3]

A. ArnoldP. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), 29 (2000), 571-581.  doi: 10.1080/00411450008205893.

[4]

D. BalaguéJ. A. CarrilloT. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.  doi: 10.1007/s00205-013-0644-6.

[5]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO-Modélisation Mathématique et Analyse Numérique, 31 (1997), 615–641. doi: 10.1051/m2an/1997310506151.

[6]

A. L. BertozziJ. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[7]

A. L. BertozziT. KolokolnikovH. SunD. Uminsky and J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985.  doi: 10.4310/CMS.2015.v13.n4.a6.

[8]

A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the newtonian potential: The dynamics of patch solutions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140005, 39pp. doi: 10.1142/S0218202511400057.

[9]

A. L. BertozziT. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Communications on Pure and Applied Mathematics, 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[10]

F. BolleyY. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107.  doi: 10.1142/S0219891605000397.

[11]

G. A. Bonaschi, Gradient flows driven by a non-smooth repulsive interaction potential, Master's thesis, University of Pavia, Italy, arXiv: 1310.3677, 2011.

[12]

G. A. BonaschiJ. A. CarrilloM. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control, Optimisation and Calculus of Variations, 21 (2015), 414-441.  doi: 10.1051/cocv/2014032.

[13]

Y. Brenier, L$^2$ formulation of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 193 (2009), 1-19.  doi: 10.1007/s00205-009-0214-0.

[14] A. Bressan, Hyperbolic Systems of Conservation Laws. The one-dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 20 edition, 2000. 
[15]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973.

[16]

M. BurgerM. Di FrancescoS. Fagioli and A. Stevens, Sorting phenomena in a mathematical model for two mutually attracting/repelling species, SIAM Journal on Mathematical Analysis, 50 (2018), 3210-3250.  doi: 10.1137/17M1125716.

[17]

M. BurgerR. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786.

[18]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.

[19]

V. CalvezJ. A. Carrillo and F. Hoffmann, The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime, Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, 2186 (2017), 1-71. 

[20]

J. CarrilloY. Huang and M. Schmidtchen, Zoology of a nonlocal cross-diffusion model for two species, SIAM Journal on Applied Mathematics, 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.

[21]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.

[22]

J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, 2 (2019), 65–108, arXiv: 1810.03634. doi: 10.1007/978-3-030-20297-2_3.

[23]

J. A. CarrilloM. G. Delgadino and A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys., 343 (2016), 747-781.  doi: 10.1007/s00220-016-2598-7.

[24]

J. A. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepcev, 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.

[25]

J. A. CarrilloL. C. Ferreira and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Advances in Mathematics, 231 (2012), 306-327.  doi: 10.1016/j.aim.2012.03.036.

[26]

J. A. Carrillo, F. Filbet and M. Schmidtchen, Convergence of a Finite Volume Scheme for a System of Interacting Species with Cross-Diffusion, arXiv e-prints, Apr. 2018.

[27]

J. A. CarrilloY. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.  doi: 10.1017/S0956792514000126.

[28]

J. A. CarrilloS. Martin and V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126.  doi: 10.1016/j.physd.2013.02.004.

[29]

J. A. Carrillo and G. Toscani, Wasserstein metric and large–time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (In Honour of the Salvatore Rionero 70th Birthday), 2005, 234–244.

[30] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. Oxford University Press, New York, 1998. 
[31]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Fourth Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 325 edition, 2016. doi: 10.1007/978-3-662-49451-6.

[32]

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.

[33]

E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions eds., Masson, 29 (1993), 81-98. 

[34]

M. Di FrancescoA. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.

[35]

M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.  doi: 10.1088/0951-7715/26/10/2777.

[36]

M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations, Calc. Var. Partial Differential Equations, 50 (2014), 199-230.  doi: 10.1007/s00526-013-0633-5.

[37]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical Review Letters, 96 (2006), 104302.

[38]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998.

[39]

J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM J. Appl. Dyn. Syst., 15 (2016), 2213-2226.  doi: 10.1137/16M1069006.

[40]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[41]

R. C. FetecauY. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[42]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker–planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[43]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011), 015203. doi: 10.1103/PhysRevE.84.015203.

[44]

S. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[45]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.

[46]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.

[47]

T.-P. Liu and M. Pierre, Source solutions and asymptotic behaviour in conservations laws, J. Differential Equations, 51 (1984), 419-441.  doi: 10.1016/0022-0396(84)90096-2.

[48]

D. MatthesR. McCann and G. Savaré, A family of fourth order equations of gradient flow type, Comm. P.D.E., 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.

[49]

R. J. McCann, A convexity principle for interacting gases, Advances in Mathematics, 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.

[50]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[51]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[52]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, volume I of Probability and Its Applications, Springer, New York, 1998.

[53]

F. Santambrogio, Optimal Transport for Applied Mathematicians, volume 86 of Progress in Nonlinear Differential Equations and Their Applications., Birkhäuser Verlag, Basel, 2015. doi: 10.1007/978-3-319-20828-2.

[54]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[55]

G. Toscani, One-dimensional kinetic models of granular flows, ESAIM: Modélisation Mathématique et Analyse Numérique, 34 (2000), 1277–1291. doi: 10.1051/m2an:2000127.

[56]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[57]

C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

[2]

L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 1-136.  doi: 10.1016/S1874-5717(07)80004-1.

[3]

A. ArnoldP. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), 29 (2000), 571-581.  doi: 10.1080/00411450008205893.

[4]

D. BalaguéJ. A. CarrilloT. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.  doi: 10.1007/s00205-013-0644-6.

[5]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO-Modélisation Mathématique et Analyse Numérique, 31 (1997), 615–641. doi: 10.1051/m2an/1997310506151.

[6]

A. L. BertozziJ. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[7]

A. L. BertozziT. KolokolnikovH. SunD. Uminsky and J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985.  doi: 10.4310/CMS.2015.v13.n4.a6.

[8]

A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the newtonian potential: The dynamics of patch solutions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140005, 39pp. doi: 10.1142/S0218202511400057.

[9]

A. L. BertozziT. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Communications on Pure and Applied Mathematics, 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[10]

F. BolleyY. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107.  doi: 10.1142/S0219891605000397.

[11]

G. A. Bonaschi, Gradient flows driven by a non-smooth repulsive interaction potential, Master's thesis, University of Pavia, Italy, arXiv: 1310.3677, 2011.

[12]

G. A. BonaschiJ. A. CarrilloM. Di Francesco and M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control, Optimisation and Calculus of Variations, 21 (2015), 414-441.  doi: 10.1051/cocv/2014032.

[13]

Y. Brenier, L$^2$ formulation of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 193 (2009), 1-19.  doi: 10.1007/s00205-009-0214-0.

[14] A. Bressan, Hyperbolic Systems of Conservation Laws. The one-dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 20 edition, 2000. 
[15]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973.

[16]

M. BurgerM. Di FrancescoS. Fagioli and A. Stevens, Sorting phenomena in a mathematical model for two mutually attracting/repelling species, SIAM Journal on Mathematical Analysis, 50 (2018), 3210-3250.  doi: 10.1137/17M1125716.

[17]

M. BurgerR. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786.

[18]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.

[19]

V. CalvezJ. A. Carrillo and F. Hoffmann, The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime, Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, 2186 (2017), 1-71. 

[20]

J. CarrilloY. Huang and M. Schmidtchen, Zoology of a nonlocal cross-diffusion model for two species, SIAM Journal on Applied Mathematics, 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.

[21]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.

[22]

J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, 2 (2019), 65–108, arXiv: 1810.03634. doi: 10.1007/978-3-030-20297-2_3.

[23]

J. A. CarrilloM. G. Delgadino and A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys., 343 (2016), 747-781.  doi: 10.1007/s00220-016-2598-7.

[24]

J. A. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepcev, 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.

[25]

J. A. CarrilloL. C. Ferreira and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Advances in Mathematics, 231 (2012), 306-327.  doi: 10.1016/j.aim.2012.03.036.

[26]

J. A. Carrillo, F. Filbet and M. Schmidtchen, Convergence of a Finite Volume Scheme for a System of Interacting Species with Cross-Diffusion, arXiv e-prints, Apr. 2018.

[27]

J. A. CarrilloY. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.  doi: 10.1017/S0956792514000126.

[28]

J. A. CarrilloS. Martin and V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126.  doi: 10.1016/j.physd.2013.02.004.

[29]

J. A. Carrillo and G. Toscani, Wasserstein metric and large–time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (In Honour of the Salvatore Rionero 70th Birthday), 2005, 234–244.

[30] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. Oxford University Press, New York, 1998. 
[31]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Fourth Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 325 edition, 2016. doi: 10.1007/978-3-662-49451-6.

[32]

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.

[33]

E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions eds., Masson, 29 (1993), 81-98. 

[34]

M. Di FrancescoA. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.

[35]

M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.  doi: 10.1088/0951-7715/26/10/2777.

[36]

M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations, Calc. Var. Partial Differential Equations, 50 (2014), 199-230.  doi: 10.1007/s00526-013-0633-5.

[37]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical Review Letters, 96 (2006), 104302.

[38]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998.

[39]

J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM J. Appl. Dyn. Syst., 15 (2016), 2213-2226.  doi: 10.1137/16M1069006.

[40]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[41]

R. C. FetecauY. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[42]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker–planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[43]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011), 015203. doi: 10.1103/PhysRevE.84.015203.

[44]

S. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[45]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.

[46]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.

[47]

T.-P. Liu and M. Pierre, Source solutions and asymptotic behaviour in conservations laws, J. Differential Equations, 51 (1984), 419-441.  doi: 10.1016/0022-0396(84)90096-2.

[48]

D. MatthesR. McCann and G. Savaré, A family of fourth order equations of gradient flow type, Comm. P.D.E., 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.

[49]

R. J. McCann, A convexity principle for interacting gases, Advances in Mathematics, 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.

[50]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[51]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[52]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, volume I of Probability and Its Applications, Springer, New York, 1998.

[53]

F. Santambrogio, Optimal Transport for Applied Mathematicians, volume 86 of Progress in Nonlinear Differential Equations and Their Applications., Birkhäuser Verlag, Basel, 2015. doi: 10.1007/978-3-319-20828-2.

[54]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[55]

G. Toscani, One-dimensional kinetic models of granular flows, ESAIM: Modélisation Mathématique et Analyse Numérique, 34 (2000), 1277–1291. doi: 10.1051/m2an:2000127.

[56]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[57]

C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

Figure 1.  This example has two separated indicator functions as initial data. In the left graph we see the evolution of system (6) to the stationary state (right graph). In the middle we see the energy (black) of the solution and its dissipation (red). The dotted line is the numerical time derivative of the energy. It matches well with the analytically obtained dissipation
Figure 2.  We choose partially overlapping initial data and observe, as before that mixing occurs. The graph on the left displays the evolution of both densities at different time instances, while the rightmost graph displays the stationary state with identical densities. The graph in the middle shows the energy decay along the solution and the numerical dissipation and the analytical dissipation agree well
Figure 3.  Initial (left) and exact solution (right) at time $ t = 0.5 $ for the case of two distinct Dirac deltas at the level of distribution functions
Figure 4.  Initial (left) and exact solution (right) at time $ t = 1/(4(1-m)) $ with $ m = 0.4 $ for the case of two partially overlapping deltas at the level of distribution functions
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