# American Institute of Mathematical Sciences

January  2021, 14(1): 89-113. doi: 10.3934/krm.2020050

## Superposition principle and schemes for measure differential equations

 1 "Sapienza" Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, via Scarpa 16, I-00161 Rome, Italy 2 Politecnico di Milano, Dipartimento di Matematica "F. Brioschi", Piazza Leonardo da Vinci 32, I-20133 Milano, Italy 3 Rutgers University - Camden, Department of Mathematical Sciences, 311 N. 5th Street, Camden, NJ 08102, USA

* Corresponding author: Giulia Cavagnari

Received  February 2020 Revised  September 2020 Published  November 2020

Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savaré, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.

Citation: Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^nd$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/b137080.  Google Scholar [2] M. Bongini and G. Buttazzo, Optimal control problems in transport dynamics, Mathematical Models and Methods in Applied Sciences, 27, (2017), 427–451. doi: 10.1142/S0218202517500063.  Google Scholar [3] F. Camilli, R. De Maio and A. Tosin, Measure-valued solutions to nonlocal transport equations on networks, J. Differential Equations, 264 (2018), 7213-7241.  doi: 10.1016/j.jde.2018.02.015.  Google Scholar [4] F. Camilli, R. De Maio and A. Tosin, Transport of measures on networks, Networks & Heterogeneous Media, 12 (2017), 191-215.  doi: 10.3934/nhm.2017008.  Google Scholar [5] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar [6] G. Cavagnari, A. Marigonda and B. Piccoli, Generalized dynamic programming principle and sparse mean-field control problems, Journal of Mathematical Analysis and Applications, 481, (2020), 123437, 45 pp. doi: 10.1016/j.jmaa.2019.123437.  Google Scholar [7] G. Cavagnari, A. Marigonda and B. Piccoli, Superposition principle for differential inclusions, in Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science (eds. I. Lirkov and S. Margenov), 10665, Springer, Cham, 2018, 201–209. doi: 10.1007/978-3-319-73441-5_21.  Google Scholar [8] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS & A: Modeling, Simulation and Applications, Springer, Cham, Vol. 12, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar [9] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977. doi: 10.1090/surv/015.  Google Scholar [10] F. Golse, The mean-field limit for the dynamics of large particle systems, Journées Équations aux Dérivées Partielles, Univ. Nantes, Nantes, 2003, 47 pp. doi: 10.5802/jedp.623.  Google Scholar [11] P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic & Related Models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661.  Google Scholar [12] C. Orrieri, Large deviations for interacting particle systems: joint mean-field and small-noise limit, Electron. J. Probab., 25 (2020) Paper No. 111, 44 pp. doi: 10.1214/20-EJP516.  Google Scholar [13] B. Piccoli, Measure differential equations, Arch Rational Mech Anal, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4.  Google Scholar [14] B. Piccoli and F. Rossi, Measure dynamics with probability vector fields and sources, Discrete & Continuous Dynamical Systems - A, 39 (2019), 6207-6230.  doi: 10.3934/dcds.2019270.  Google Scholar [15] B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Applicandae Mathematicae, 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.  Google Scholar [16] F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Communications in Partial Differential Equations, 22, (1997), 225–267. doi: 10.1080/03605309708821265.  Google Scholar [17] F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Basel, vol. 87, ed. 1 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar [18] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar [19] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^nd$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/b137080.  Google Scholar [2] M. Bongini and G. Buttazzo, Optimal control problems in transport dynamics, Mathematical Models and Methods in Applied Sciences, 27, (2017), 427–451. doi: 10.1142/S0218202517500063.  Google Scholar [3] F. Camilli, R. De Maio and A. Tosin, Measure-valued solutions to nonlocal transport equations on networks, J. Differential Equations, 264 (2018), 7213-7241.  doi: 10.1016/j.jde.2018.02.015.  Google Scholar [4] F. Camilli, R. De Maio and A. Tosin, Transport of measures on networks, Networks & Heterogeneous Media, 12 (2017), 191-215.  doi: 10.3934/nhm.2017008.  Google Scholar [5] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar [6] G. Cavagnari, A. Marigonda and B. Piccoli, Generalized dynamic programming principle and sparse mean-field control problems, Journal of Mathematical Analysis and Applications, 481, (2020), 123437, 45 pp. doi: 10.1016/j.jmaa.2019.123437.  Google Scholar [7] G. Cavagnari, A. Marigonda and B. Piccoli, Superposition principle for differential inclusions, in Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science (eds. I. Lirkov and S. Margenov), 10665, Springer, Cham, 2018, 201–209. doi: 10.1007/978-3-319-73441-5_21.  Google Scholar [8] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS & A: Modeling, Simulation and Applications, Springer, Cham, Vol. 12, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar [9] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977. doi: 10.1090/surv/015.  Google Scholar [10] F. Golse, The mean-field limit for the dynamics of large particle systems, Journées Équations aux Dérivées Partielles, Univ. Nantes, Nantes, 2003, 47 pp. doi: 10.5802/jedp.623.  Google Scholar [11] P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic & Related Models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661.  Google Scholar [12] C. Orrieri, Large deviations for interacting particle systems: joint mean-field and small-noise limit, Electron. J. Probab., 25 (2020) Paper No. 111, 44 pp. doi: 10.1214/20-EJP516.  Google Scholar [13] B. Piccoli, Measure differential equations, Arch Rational Mech Anal, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4.  Google Scholar [14] B. Piccoli and F. Rossi, Measure dynamics with probability vector fields and sources, Discrete & Continuous Dynamical Systems - A, 39 (2019), 6207-6230.  doi: 10.3934/dcds.2019270.  Google Scholar [15] B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes, Acta Applicandae Mathematicae, 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.  Google Scholar [16] F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Communications in Partial Differential Equations, 22, (1997), 225–267. doi: 10.1080/03605309708821265.  Google Scholar [17] F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Basel, vol. 87, ed. 1 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar [18] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar [19] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar
LAS and semi-discrete Lagrangian schemes on the left for $N = 1$. Mean-velocity scheme on the right
LAS and semi-discrete Lagrangian schemes on the left. Mean-velocity scheme on the right
LAS scheme: for N = 1 (left) and N=2 (right)
Semi-discrete Lagrangian scheme on the left. Mean-velocity scheme on the right
Peano's brush referred to Example 4
LAS scheme for N = 1, 2, 3
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