The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations
Marsden, J. E. and S. Shkoller
Arch. Rational Mech. Anal. 166, 27-46.
The purpose of this paper is twofold. First, we give a derivation of the Lagrangian
averaged Euler (LAE-
) and Navier-Stokes (LANS-
) equations. This theory
involves a spatial scale
and the equations are designed to accurately capture the
dynamics of the Euler and Navier-Stokes equations at length scales larger than
while averaging the motion at scales smaller than
. The derivation involves an
averaging procedure that combines ideas from both
the material (Lagrangian) and
spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I.
Taylor's "frozen turbulence" hypothesis as the foundation for the model equations;
more precisely, the derivation is based on the strong physical assumption that fluctutations
are frozen into the mean flow. In this article, we use this hypothesis to derive the
averaged Lagrangian for the theory, and all the terms up to and including order
are accounted for.
The equations come in both an isotropic and anisotropic
version. The anisotropic equations are a coupled system of PDEs (partial differential
equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works
by FOIAS, HOLM & TITI , and ourselves , an analysis of the isotropic
equations has been given. In the second part of this paper, we establish local in time
well-posedness of the anisotropic LANS- equations using quasilinear PDE type methods.