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The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

Marsden, J. E. and Ratiu, T. S. and Shkoller, S. (2000) The geometry and analysis of the averaged Euler equations and a new diffeomorphism group. Geometric and Functional Analysis, 10 (3). pp. 582-599. ISSN 1016-443X.

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This paper develops the geometric analysis of geodesic flow of a new right invariant metric {.,.}_1 on two subgroups of the volume preserving diffeomorphism group of a smooth n-dimensional compact subset Ω of R^n with C^∞ boundary ∂Ω . The geodesic equations are given by the system of PDEs v_(t) + ∇_(u(t))v(t)−є[∇u(t)]^t•Δu(t) = −grad p(t) in Ω, v = (1−єΔ)u; div u = 0; u(0) = u_0; which are the averaged Euler (or Euler-α) equations when є = α2 is a length scale, and are the equations of an inviscid non-newtonian second grade fluid when є = α1, a material parameter. The boundary conditions associated with the geodesic flow on the two groups we study are given by either u = 0 on ∂Ω or u • n = 0 and (∇nu)^(tan) + S_n(u) = 0 on ∂Ω where n is the outward pointing unit normal on ∂Ω, and where S_n is the second fundamental form of ∂Ω. We prove that for initial data u_0 in H^s, s > (n/2) + 1, the above system of PDE are well-posed, by establishing existence, uniqueness, and smoothness of the geodesic spray of the metric {.,.}_1, together with smooth dependence on initial data. We are then able to prove that the limit of zero viscosity for the corresponding viscous equations is a regular limit.

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Ratiu, T. S.0000-0003-1972-5768
Additional Information:© Birkhäuser Verlag, Basel 2000. Submitted: November 1998. Revision: February 1999. Final version: July 1999. The authors would like to thank the anonymous referee for many helpful suggestions that improved the manuscript. J.E.M. and S.S. were partially supported by an NSF-KDI grant ATM-98-73133, and T.S.R. was partially supported by NSF grant DMS-98-02378 and the Swiss NSF. S.S. would also like to thank the Center for Nonlinear Studies in Los Alamos for providing an excellent environment wherein much of this work was performed.
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Swiss National Science Foundation (SNSF)UNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20100907-142004839
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19808
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 15:21
Last Modified:03 Oct 2019 02:01

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