The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits

Claude Bardos; Nicolas Besse

doi:10.3934/krm.2013.6.893

Article Contents

2013, Volume 6, Issue 4: 893-917. Doi: 10.3934/krm.2013.6.893

This issue Previous Article Energy estimate for a linear symmetric hyperbolic-parabolic system in half line Next Article Empirical measures and Vlasov hierarchies

The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits

Claude Bardos^1, and
Nicolas Besse^2,

1.
Laboratoire Jacques Louis Lions UMR CNRS 7598, Université Denis Diderot et Université Pierre et, Marie Curie 2 Place Jussieu 75005 Paris
2.
Institut Jean Lamour UMR CNRS 7198, Université de Lorraine BP 70239 54506, Vandoeuvre-lès-Nancy Cedex

Received: July 31, 2013

Revised: August 31, 2013

Published: November 2013

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Abstract

This contribution concerns a one-dimensional version of the Vlasov equation dubbed the Vlasov$-$Dirac$-$Benney equation (in short V$-$D$-$B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with the so-called Benney equation leads to new stability results. Eventually the V$-$D$-$B appears to be at the ``cross road" of several problems of mathematical physics which have as far as stability is concerned very similar properties.

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Mathematics Subject Classification: Primary: 3Q83, 75X05; Secondary: 82D10.

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