# American Institute of Mathematical Sciences

December  2013, 6(4): 893-917. doi: 10.3934/krm.2013.6.893

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

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

Received  August 2013 Revised  September 2013 Published  November 2013

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.
Citation: Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893
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