# American Institute of Mathematical Sciences

November  2019, 39(11): 6207-6230. doi: 10.3934/dcds.2019270

## Measure dynamics with Probability Vector Fields and sources

 1 Department of Mathematical Sciences, Rutgers University - Camden, Camden, NJ, USA 2 Dipartimento di Matematica "Tullio Levi–Civita", Università degli Studi di Padova, Padova, Italy

* Corresponding author: Francesco Rossi

Received  September 2018 Revised  April 2019 Published  August 2019

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass.

The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

Citation: Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270
##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2. Google Scholar [2] L. Ambrosio, M. Colombo and A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields, Archive for Rational Mechanics and Analysis, 218 (2015), 1043-1081. doi: 10.1007/s00205-015-0875-9. Google Scholar [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. Google Scholar [5] J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156. doi: 10.1007/s10440-012-9758-3. Google Scholar [6] J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003. Google Scholar [7] J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction cross-diffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), 5695-5718. doi: 10.1137/17M1158379. Google Scholar [8] L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, An interpolating distance between optimal transport and Fisher-Rao metrics, Foundations of Computational Mathematics, 18 (2018), 1-44. doi: 10.1007/s10208-016-9331-y. Google Scholar [9] G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Tesi. Scuola Normale Superiore di Pisa (Nuova Series) [Theses of Scuola Normale Superiore di Pisa (New Series)], 12. Edizioni della Normale, Pisa, 2009. Google Scholar [10] G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. Google Scholar [11] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. Google Scholar [12] R. M. Dudley, Real Analysis and Probability, 2nd edition, Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, 2002. doi: 10.1017/CBO9780511755347. Google Scholar [13] J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097. doi: 10.1016/j.jde.2015.02.037. Google Scholar [14] J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal on Mathematical Analysis, 48 (2016), 1929-1953. doi: 10.1137/15M1031655. Google Scholar [15] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. Google Scholar [16] P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773. doi: 10.1142/S021989161000227X. Google Scholar [17] L. G. Hanin, Kantorovich-Rubinstein norm and its application in the theory of lipschitz spaces, Proceedings of the American Mathematical Society, 115 (1992), 345-352. doi: 10.2307/2159251. Google Scholar [18] S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differential Equations, 21 (2016), 1117-1164. Google Scholar [19] M. Liero, A. Mielke and G. Savaré, Optimal transport in competition with reaction: The Hellinger–Kantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), 2869-2911. doi: 10.1137/15M1041420. Google Scholar [20] M. Liero, A. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Inventiones Mathematicae, 211 (2018), 969-1117. doi: 10.1007/s00222-017-0759-8. Google Scholar [21] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation, Springer, 1048 (1984), 60–110. doi: 10.1007/BFb0071878. Google Scholar [22] B. Piccoli, Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317. doi: 10.1007/s00205-019-01379-4. Google Scholar [23] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358. doi: 10.1007/s00205-013-0669-x. Google Scholar [24] B. Piccoli and F. Rossi, On properties of the generalized wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365. doi: 10.1007/s00205-016-1026-7. Google Scholar [25] A. Ulikowska, An age-structured two-sex model in the space of Radon measures: Well posedness, Kinetic and Related Models, 5 (2012), 873-900. doi: 10.3934/krm.2012.5.873. Google Scholar [26] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar [27] C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

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##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2. Google Scholar [2] L. Ambrosio, M. Colombo and A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields, Archive for Rational Mechanics and Analysis, 218 (2015), 1043-1081. doi: 10.1007/s00205-015-0875-9. Google Scholar [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. Google Scholar [5] J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156. doi: 10.1007/s10440-012-9758-3. Google Scholar [6] J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003. Google Scholar [7] J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction cross-diffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), 5695-5718. doi: 10.1137/17M1158379. Google Scholar [8] L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, An interpolating distance between optimal transport and Fisher-Rao metrics, Foundations of Computational Mathematics, 18 (2018), 1-44. doi: 10.1007/s10208-016-9331-y. Google Scholar [9] G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Tesi. Scuola Normale Superiore di Pisa (Nuova Series) [Theses of Scuola Normale Superiore di Pisa (New Series)], 12. Edizioni della Normale, Pisa, 2009. Google Scholar [10] G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. Google Scholar [11] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. Google Scholar [12] R. M. Dudley, Real Analysis and Probability, 2nd edition, Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, 2002. doi: 10.1017/CBO9780511755347. Google Scholar [13] J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097. doi: 10.1016/j.jde.2015.02.037. Google Scholar [14] J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal on Mathematical Analysis, 48 (2016), 1929-1953. doi: 10.1137/15M1031655. Google Scholar [15] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. Google Scholar [16] P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773. doi: 10.1142/S021989161000227X. Google Scholar [17] L. G. Hanin, Kantorovich-Rubinstein norm and its application in the theory of lipschitz spaces, Proceedings of the American Mathematical Society, 115 (1992), 345-352. doi: 10.2307/2159251. Google Scholar [18] S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differential Equations, 21 (2016), 1117-1164. Google Scholar [19] M. Liero, A. Mielke and G. Savaré, Optimal transport in competition with reaction: The Hellinger–Kantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), 2869-2911. doi: 10.1137/15M1041420. Google Scholar [20] M. Liero, A. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Inventiones Mathematicae, 211 (2018), 969-1117. doi: 10.1007/s00222-017-0759-8. Google Scholar [21] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation, Springer, 1048 (1984), 60–110. doi: 10.1007/BFb0071878. Google Scholar [22] B. Piccoli, Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317. doi: 10.1007/s00205-019-01379-4. Google Scholar [23] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358. doi: 10.1007/s00205-013-0669-x. Google Scholar [24] B. Piccoli and F. Rossi, On properties of the generalized wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365. doi: 10.1007/s00205-016-1026-7. Google Scholar [25] A. Ulikowska, An age-structured two-sex model in the space of Radon measures: Well posedness, Kinetic and Related Models, 5 (2012), 873-900. doi: 10.3934/krm.2012.5.873. Google Scholar [26] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. Google Scholar [27] C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar
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