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

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

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