Generalized vorticity for bubbly liquid and dispersive shallow water equations

The aim of this article is to study the main properties for a class of lagrangian models describing, in particular, bubbly liquids and dispersive shallow water flows. Important notions of generalized vorticity and generalized potential flows are introduced which permits one to prove the analogues of classical theorems of ideal Fluid Mechanics: Lagrange, Cauchy, Kelvin and Bernoulli theorems. A non-local Hamiltonian formulation of the generalized potential flows is obtained. A generalization of the classical singular solutions: 2-D vortex-source, axisymmetric swirl, and solutions with a uniform space distribution of the pressure have also been found.