The Cuachy problem for the 3D vortical flow of ideal barotropic fluid is considered. It is shown that the solution of the Cauchy problem is unique, if one considers seven dynamic equations for seven dependent variables: the density ρ, the velocity v and Lagrangian variables ξ = {ξ 1 , ξ 2 , ξ 3 }, labeling the fluid particles. If one considers only the closed (Euler) system of four equations fo...

In this paper, we study the convergence of time-dependent Euler–Poisson equations to incompressible type Euler equations via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and the symmetric hyperbolic property of the systems are used to justify the convergence of the limit.

We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of Rn or an n-dimensional manifold M). Our main focus will be on the case of quadratic Lagrangians; that is, on geodesic motion on the diffeomorphism group with respect to the right invariant Sobolev H metric. The correspondi...

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.