Hamiltonian Structure, Fluid Representation and Stability for the Vlasov–Dirac–Benney Equation

  • Claude BardosEmail author
  • Nicolas Besse
Part of the Fields Institute Communications book series (FIC, volume 75)


This contribution is an element of a research program devoted to the analysis of a variant of the Vlasov–Poisson equation that we dubbed the Vlasov–Dirac–Benney equation or in short V–D–B equation. As such it contains both new results and efforts to synthesize previous observations. One of main links between the different issues is the use of the energy of the system. In some cases such energy becomes a convex functional and allows to extend to the present problem the methods used in the study of conservation laws. Such use of the energy is closely related to the Hamiltonian structure of the problem. Hence it is a pleasure to present this article to Walter Craig in recognition to the pioneering work he made for our community, among other things, on the relations between Hamiltonian systems and Partial Differential Equations.


Cauchy Problem Hamiltonian System Poisson Equation Vlasov Equation Hamiltonian Structure 
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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Denis DiderotParisFrance
  2. 2.Département Physique de la Matière et des MatériauxInstitut Jean Lamour UMR CNRS 7198, Université de LorraineVandoeuvre-lès-Nancy CedexFrance

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