A Poisson map from kinetic theory to hydrodynamics with non-constant entropy

Variable-moment fluid closures with Hamiltonian structure

“…The fact that the map ( 13 ) is Poisson justifies the claim from " Poisson bracket on the space of truncated fluid moment vectors " that the bracket repairs the shortcoming of the Hamiltonian fluid moment closures due to Holm and Tronci 25 that do not include the density moment. It is interesting to note that Chong 44 previously constructed a Poisson map that sends phase space distributions into fluid-like quantities; that Poisson map appears to be unrelated to this one.…”

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

“…It is appropriate to conclude this subsection by mentioning some works that are related to the spirit of our paper in terms of understanding the role of the Hamiltonian formulation of PDE in mathematical physics. We first mention some recent work of Chong [Cho22] which exhibits a Poisson map from the Poisson manifold underlying the Vlasov equation to the Poisson manifold underlying the compressible Euler equation. We also mention impressive work of Khesin et al [KMeM19, KMeM19, KMeM21] which shows that the Madelung transformation from wave functions to hydrodynamic variables is a Kähler morphism and which develops a geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities, covering a number of equations, including (in)compressible fluid and (non)linear Schrödinger equations.…”