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