## Abstract

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that L ^{1} spaces are not natural for such equations, since we lose uniqueness of the solution.

Original language | English (US) |
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Pages (from-to) | 73-105 |

Number of pages | 33 |

Journal | Acta Applicandae Mathematicae |

Volume | 124 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2013 |

## All Science Journal Classification (ASJC) codes

- Applied Mathematics

## Keywords

- Evolution of measures Wasserstein distance
- Numerical schemes for PDEs
- Pedestrian modelling
- Transport equation