989 research outputs found
On Global Well-Posedness of the Lagrangian Averaged Euler Equations
We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the bounded mean oscillation of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we obtain the global existence of the averaged two-dimensional (2D) Boussinesq equations and the Lagrangian averaged 2D quasi-geostrophic equations in finite Sobolev space in the absence of viscosity or dissipation
The anisotropic averaged Euler equations
The purpose of this paper is to derive the anisotropic averaged Euler
equations and to study their geometric and analytic properties. These new
equations involve the evolution of a mean velocity field and an advected
symmetric tensor that captures the fluctuation effects. Besides the derivation
of these equations, the new results in the paper are smoothness properties of
the equations in material representation, which gives well-posedness of the
equations, and the derivation of a corrector to the macroscopic velocity field.
The numerical implementation and physical implications of this set of equations
will be explored in other publications.Comment: 24 pages, 1 figur
The geometry and analysis of the averaged Euler equations and a new diffeomorphism group
We present a geometric analysis of the incompressible averaged Euler
equations for an ideal inviscid fluid. We show that solutions of these
equations are geodesics on the volume-preserving diffeomorphism group of a new
weak right invariant pseudo metric. We prove that for precompact open subsets
of , this system of PDEs with Dirichlet boundary conditions are
well-posed for initial data in the Hilbert space , . We then use
a nonlinear Trotter product formula to prove that solutions of the averaged
Euler equations are a regular limit of solutions to the averaged Navier-Stokes
equations in the limit of zero viscosity. This system of PDEs is also the model
for second-grade non-Newtonian fluids
The vortex blob method as a second-grade non-Newtonian fluid
We show that a certain class of vortex blob approximations for ideal
hydrodynamics in two dimensions can be rigorously understood as solutions to
the equations of second-grade non-Newtonian fluids with zero viscosity, and
initial data in the space of Radon measures . The
solutions of this regularized PDE, also known as the averaged Euler or
Euler- equations, are geodesics on the volume preserving diffeomorphism
group with respect to a new weak right invariant metric. We prove global
existence of unique weak solutions (geodesics) for initial vorticity in
such as point-vortex data, and show that the
associated coadjoint orbit is preserved by the flow. Moreover, solutions of
this particular vortex blob method converge to solutions of the Euler equations
with bounded initial vorticity, provided that the initial data is approximated
weakly in measure, and the total variation of the approximation also converges.
In particular, this includes grid-based approximation schemes of the type that
are usually used for vortex methods
A Nonlinear Analysis of the Averaged Euler Equations
This paper develops the geometry and analysis of the averaged Euler equations
for ideal incompressible flow in domains in Euclidean space and on Riemannian
manifolds, possibly with boundary. The averaged Euler equations involve a
parameter ; one interpretation is that they are obtained by ensemble
averaging the Euler equations in Lagrangian representation over rapid
fluctuations whose amplitudes are of order . The particle flows
associated with these equations are shown to be geodesics on a suitable group
of volume preserving diffeomorphisms, just as with the Euler equations
themselves (according to Arnold's theorem), but with respect to a right
invariant metric instead of the metric. The equations are also
equivalent to those for a certain second grade fluid. Additional properties of
the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden
theorem) are also shown to hold. Using this nonlinear analysis framework, the
limit of zero viscosity for the corresponding viscous equations is shown to be
a regular limit, {\it even in the presence of boundaries}.Comment: 25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of
his 60th birthday, Arnold Festschrift Volume 2 (in press
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