On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs

Ammari Zied; Liard Quentin

doi:10.3934/dcds.2018032

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2018, Volume 38, Issue 2: 723-748. Doi: 10.3934/dcds.2018032

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On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs

Ammari Zied^1, , and
Liard Quentin^2,

1.
IRMAR, UMR-CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France
2.
LAGA, UMR-CNRS 9345, Université de Paris 13, av. J. B. Clément, 93430 Villetaneuse, France

* Corresponding author: Ammari Zied
* Corresponding author: Ammari Zied

Received: September 30, 2016

Revised: July 31, 2017

Published: November 2017

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In this paper, the Cauchy problem of classical Hamiltonian PDEs is recast into a Liouville's equation with measure-valued solutions. Then a uniqueness property for the latter equation is proved under some natural assumptions. Our result extends the method of characteristics to Hamiltonian systems with infinite degrees of freedom and it applies to a large variety of Hamiltonian PDEs (Hartree, Klein-Gordon, Schrödinger, Wave, Yukawa $\dots$). The main arguments in the proof are a projective point of view and a probabilistic representation of measure-valued solutions to continuity equations in finite dimension.

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Mathematics Subject Classification: Primary: 35Q82, 35A02, 35Q55; Secondary: 35Q61, 37K05, 28A33.

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