In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli’s function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value, then there exists a unique global subsonic solution in a suitable class for a ge...

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geo...

These notes describe how to do a piecewise linear or piecewise parabolic method for the Euler equations. 1 Euler equation properties The Euler equations in one dimension appear as: ∂ρ ∂t + ∂(ρu) ∂x = 0 (1) ∂(ρu) ∂t + ∂(ρuu + p) ∂x = 0 (2) ∂(ρE) ∂t + ∂(ρuE + up) ∂x = 0 (3) These represent conservation of mass, momentum, and energy. Here ρ is the density, u is the one-dimensional velocity, p is t...

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincar e) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the un-perturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of 1 this dissipation mech...

Given a Lagrangian L : J 1 P → R, with P = M × G → M, invariant under the natural action of G on J 1 P, we deduce the analog of the Euler–Poincaré equations. The geometry of the reduced variational problem as well as its link with the Noether Theorem and an example are also given.

We prove that the 3-D compressible Euler equations with surface tension along the moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [8] as p(ρ) = αρ − β for consants γ > 1 and α, β > 0. The analysis is made difficult by two competing nonlinearities associated with the potential...

A novel procedure to reduce by four the order of Euler–Lagrange equations associated to n-th order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two C∞symmetries is established. The method is based on a pair of variational C∞-symmetries whose commutators satisfy a certain solvability condition. It allows one to recover...

Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for classical topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of se...

This is the rst of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the pro...