Superposition principle and schemes for measure differential equations

Fabio Camilli; Giulia Cavagnari; Raul De Maio; Benedetto Piccoli

doi:10.3934/krm.2020050

Article Contents

2021, Volume 14, Issue 1: 89-113. Doi: 10.3934/krm.2020050

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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
^* Corresponding author: Giulia Cavagnari

Received: February 2020

Revised: September 2020

Early access: November 2020

Published: February 2021

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Abstract

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.

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Mathematics Subject Classification: Primary:35S99, 28A50;Secondary:35F20, 35F25.

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Figure 1. LAS and semi-discrete Lagrangian schemes on the left for $ N = 1 $. Mean-velocity scheme on the right

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Figure 2. LAS and semi-discrete Lagrangian schemes on the left. Mean-velocity scheme on the right

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Figure 3. LAS scheme: for N = 1 (left) and N=2 (right)

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Figure 4. Semi-discrete Lagrangian scheme on the left. Mean-velocity scheme on the right

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Figure 5. Peano's brush referred to Example 4

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Figure 6. LAS scheme for N = 1, 2, 3

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