title: Hamiltonian formalism of two-dimensional Vlasov kinetic equation.
abstract: In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.
authors.affiliation: Sector of Mathematical Physics, Lebedev Physical Institute of Russian Academy of Sciences , Leninskij Prospekt 53, 119991 Moscow, Russian Federation ; Department of Applied Mathematics , National Research Nuclear University MEPHI , 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation.
authors.forename: Maxim V
authors.initials: MV
authors.lastname: Pavlov
keywords: Hamiltonian structure; collisionless kinetic equation; hydrodynamic lattice; hydrodynamic reduction