# American Institute of Mathematical Sciences

February  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  February 2021 Early access  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.

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Citation: Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic and 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. [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. [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. [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. [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. [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. [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. [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. [9] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977. doi: 10.1090/surv/015. [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. [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. [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. [13] B. Piccoli, Measure differential equations, Arch Rational Mech Anal, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4. [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. [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. [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. [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. [18] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004. [19] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.

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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. [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. [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. [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. [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. [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. [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. [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. [9] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977. doi: 10.1090/surv/015. [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. [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. [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. [13] B. Piccoli, Measure differential equations, Arch Rational Mech Anal, 233 (2019), 1289-1317.  doi: 10.1007/s00205-019-01379-4. [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. [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. [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. [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. [18] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004. [19] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.
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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