Viscous Aubry-Mather theory and the Vlasov equation

Ugo Bessi

doi:10.3934/dcds.2014.34.379

Article Contents

2014, Volume 34, Issue 2: 379-420. Doi: 10.3934/dcds.2014.34.379

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Viscous Aubry-Mather theory and the Vlasov equation

Ugo Bessi^1,

1.
Dip. di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Roma

Received: October 31, 2012

Revised: April 30, 2013

Published: January 2014

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Abstract

The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.

Keywords:

Viscous Aubry-Mather theory,
Vlasov equation.

Mathematics Subject Classification: Primary: 70H20; Secondary: 35Q83.

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