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The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations

Marsden, Jerrold E. and Shkoller, Steve (2003) The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations. Archive for Rational Mechanics and Analysis, 166 (1). pp. 27-46. ISSN 0003-9527. doi:10.1007/s00205-002-0207-8.

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The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. 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 α^2 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 [10], and ourselves [16], 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-α equations using quasilinear PDE type methods.

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Additional Information:© 2002 Springer-Verlag. Accepted September 2, 2002. Published online November 26, 2002. Dedicated to Stuart Antman on the occasion of his 60th birthday. Communicated by S. Müller. We thank Marcel Oliver for various helpful and important comments on the derivation, and Daniel Coutand for a number of helpful comments about the manuscript. We would also like to thank Ciprian Foias, Darryl Holm, and Edriss Titi for many useful conversations on this topic. Finally, we would like to express our gratitude to the editor for his assistance in reading and correcting some of the finer points of the manuscript. JEM and SS were partially supported by the NSF-KDI grantATM-98-73133. JEM also acknowledges the support of the California Institute of Technology. SS was also partially supported by NSF DMS-0105004 and the Alfred P. Sloan Foundation Research Fellowship.
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Alfred P. Sloan FoundationUNSPECIFIED
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Record Number:CaltechAUTHORS:20100909-114244082
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Official Citation:MARSDEN, J., SHKOLLER, S. The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations. Arch. Rational Mech. Anal. 166, 27–46 (2003).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19848
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 22:54
Last Modified:08 Nov 2021 23:55

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