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

José Antonio Carrillo; Marco Di Francesco; Antonio Esposito; Simone Fagioli; Markus Schmidtchen

doi:10.3934/dcds.2020075

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

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

José Antonio Carrillo ^1,, , Marco Di Francesco ^2, , Antonio Esposito ^2, , Simone Fagioli ^2, and Markus Schmidtchen ^1,

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.

Keywords: Systems of aggregation equations, Newtonian potentials, uniqueness of solutions, gradient flows, long time asymptotics.

Mathematics Subject Classification: Primary: 45K05, 35A15; Secondary: 92D25, 35L45, 35L80.

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. Google Scholar
[3]	A. Arnold, P. 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. Google Scholar
[4]	D. Balagué, J. A. Carrillo, T. 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. Google Scholar
[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. Google Scholar
[6]	A. L. Bertozzi, J. 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. Google Scholar
[7]	A. L. Bertozzi, T. Kolokolnikov, H. Sun, D. 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. Google Scholar
[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. Google Scholar
[9]	A. L. Bertozzi, T. 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. Google Scholar
[10]	F. Bolley, Y. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107. doi: 10.1142/S0219891605000397. Google Scholar
[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. 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, Optimisation and Calculus of Variations, 21 (2015), 414-441. doi: 10.1051/cocv/2014032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973. Google Scholar
[16]	M. Burger, M. Di Francesco, S. 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. Google Scholar
[17]	M. Burger, R. 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. Google Scholar
[18]	V. Calvez, J. 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. Google Scholar
[19]	V. Calvez, J. 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. Google Scholar
[20]	J. Carrillo, Y. 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. Google Scholar
[21]	J. A. Carrillo, A. 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. Google Scholar
[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. Google Scholar
[23]	J. A. Carrillo, M. 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. Google Scholar
[24]	J. A. Carrillo, M. Di Francesco, A. Figalli, T. 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. Google Scholar
[25]	J. A. Carrillo, L. 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. Google Scholar
[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. Google Scholar
[27]	J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578. doi: 10.1017/S0956792514000126. Google Scholar
[28]	J. A. Carrillo, S. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	M. Di Francesco, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar
[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. Google Scholar
[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. Google Scholar
[41]	R. C. Fetecau, Y. 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. Google Scholar
[42]	R. Jordan, D. 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. Google Scholar
[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. Google Scholar
[44]	S. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	D. Matthes, R. 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. Google Scholar
[49]	R. J. McCann, A convexity principle for interacting gases, Advances in Mathematics, 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, volume I of Probability and Its Applications, Springer, New York, 1998. Google Scholar
[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. Google Scholar
[54]	C. M. Topaz, A. 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. Google Scholar
[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. Google Scholar
[56]	C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar
[57]	C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	A. Arnold, P. 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. Google Scholar
[4]	D. Balagué, J. A. Carrillo, T. 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. Google Scholar
[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. Google Scholar
[6]	A. L. Bertozzi, J. 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. Google Scholar
[7]	A. L. Bertozzi, T. Kolokolnikov, H. Sun, D. 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. Google Scholar
[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. Google Scholar
[9]	A. L. Bertozzi, T. 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. Google Scholar
[10]	F. Bolley, Y. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107. doi: 10.1142/S0219891605000397. Google Scholar
[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. 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, Optimisation and Calculus of Variations, 21 (2015), 414-441. doi: 10.1051/cocv/2014032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973. Google Scholar
[16]	M. Burger, M. Di Francesco, S. 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. Google Scholar
[17]	M. Burger, R. 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. Google Scholar
[18]	V. Calvez, J. 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. Google Scholar
[19]	V. Calvez, J. 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. Google Scholar
[20]	J. Carrillo, Y. 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. Google Scholar
[21]	J. A. Carrillo, A. 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. Google Scholar
[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. Google Scholar
[23]	J. A. Carrillo, M. 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. Google Scholar
[24]	J. A. Carrillo, M. Di Francesco, A. Figalli, T. 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. Google Scholar
[25]	J. A. Carrillo, L. 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. Google Scholar
[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. Google Scholar
[27]	J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578. doi: 10.1017/S0956792514000126. Google Scholar
[28]	J. A. Carrillo, S. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	M. Di Francesco, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar
[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. Google Scholar
[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. Google Scholar
[41]	R. C. Fetecau, Y. 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. Google Scholar
[42]	R. Jordan, D. 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. Google Scholar
[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. Google Scholar
[44]	S. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	D. Matthes, R. 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. Google Scholar
[49]	R. J. McCann, A convexity principle for interacting gases, Advances in Mathematics, 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, volume I of Probability and Its Applications, Springer, New York, 1998. Google Scholar
[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. Google Scholar
[54]	C. M. Topaz, A. 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. Google Scholar
[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. Google Scholar
[56]	C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar
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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

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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

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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

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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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American Institute of Mathematical Sciences

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

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

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