Measure dynamics with Probability Vector Fields and sources

Benedetto Piccoli; Francesco Rossi

doi:10.3934/dcds.2019270

Article Contents

2019, Volume 39, Issue 11: 6207-6230. Doi: 10.3934/dcds.2019270

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Measure dynamics with Probability Vector Fields and sources

Benedetto Piccoli^1, and
Francesco Rossi^2, ,

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
^* Corresponding author: Francesco Rossi

Received: September 2018

Revised: April 2019

Published: August 2019

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Abstract

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.

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Mathematics Subject Classification: 35S99, 35F20, 35F25.

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References

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