Abstract
In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of this distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with a source, in which both the vector field and the source depend on the measure itself. We prove the existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance.
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Ambrosio L., Gangbo W.: Hamiltonian ODEs in the Wasserstein space of probability measures. Commun. Pure Appl. Math. 61(1), 18–53 (2008)
Ambrosio L., Gigli N., Savaré G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics Eth Zurich. Birkhäuser, Basel (2008)
Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. AIMS Series on Applied Mathematics. AIMS, Springfield, (2007)
Carrillo J.A., Colombo R.M., Gwiazda P., Ulikowska A.: Structured populations, cell growth and measure valued balance laws. J. Differ. Equ. 252, 3245–3277 (2012)
Crippa G., Lecureux-Mercier M.: Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow. Nonlinear Differ. Equ. Appl. NoDEA 20(3), 523–537 (2013)
Cristiani E., Piccoli B., Tosin A.: Multiscale Modeling of Granular Flows with Application to Crowd Dynamics. Multiscale Model. Simul. 9, 155–182 (2011)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Evers J., Muntean A.: Modeling Micro-Macro Pedestrian Counterflow in Heterogeneous Domains. Nonlinear Phenom. Complex Syst. 14(1), 27–37 (2011)
Figalli A., Gigli N.: A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. Journal de Mathématiques Pures et Appliquées 94(2), 107–130 (2010)
Gwiazda P., Lorenz T., Marciniak-Czochra A.: A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients. J. Differ. Equ. 248, 2703–2735 (2010)
Gwiazda P., Marciniak-Czochra A.: Structured population equations in metric spaces. J. Hyperbolic Differ. Equ. 7(4), 733–773 (2010)
Maury B., Roudneff-Chupin A., Santambrogio F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010)
Maury B., Roudneff-Chupin A., Santambrogio F., Venel J.: Handling congestion in crowd motion modeling. Netw. Heterog. Media 6(3), 485–519 (2011)
Piccoli B., Rossi F.: Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes. Acta Applicandae Mathematicae 124, 73–105 (2013)
Piccoli, B., Rossi, F.: On properties of the Generalized Wasserstein distance, arXiv:1304.7014 (submitted)
Piccoli B., Tosin A.: Time-evolving measures and macroscopic modeling of pedestrian flow. Arch. Ration. Mech. Anal. 199(3), 707–738 (2011)
Piccoli B., Tosin A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21(2), 85–107 (2009)
Tosin A., Frasca P.: Existence and approximation of probability measure solutions to models of collective behaviors. Netw. Heterog. Media 6(3), 561–596 (2011)
Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, 2008
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI, 2003
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Piccoli, B., Rossi, F. Generalized Wasserstein Distance and its Application to Transport Equations with Source. Arch Rational Mech Anal 211, 335–358 (2014). https://doi.org/10.1007/s00205-013-0669-x
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DOI: https://doi.org/10.1007/s00205-013-0669-x