Geometric & Functional Analysis GAFA

, Volume 10, Issue 3, pp 582–599

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

  • J.E. Marsden
  • T.S. Ratiu
  • S. Shkoller

DOI: 10.1007/PL00001631

Cite this article as:
Marsden, J., Ratiu, T. & Shkoller, S. GAFA, Geom. funct. anal. (2000) 10: 582. doi:10.1007/PL00001631

Abstract.

This paper develops the geometric analysis of geodesic ow of a new right invariant metric \( \langle \cdot,\cdot \rangle_1 \) on two subgroups of the volume preserving diffeomorphism group of a smooth n-dimensional compact subset \( \Omega \) of \( {\Bbb R}^2 \) with \( C^{\infty} \) boundary \( \partial \Omega \). The geodesic equations are given by the system of PDEs¶¶\( {\dot v}(t) + \nabla_{u(t)}v(t) - \epsilon[\nabla u(t)]^{t} \cdot \triangle u(t) = -\,{\rm grad}\,p(t)\,{\rm in}\,\Omega \)\( v = (1 - \epsilon\triangle)u,\qquad {\rm div}\,u = 0 \)u(0) = u0,¶which are the averaged Euler (or Euler-\( \alpha \)) equations when \( \epsilon = \alpha^2 \) is a length scale, and are the equations of an inviscid non-newtonian second grade uid when \( \epsilon = \tilde \alpha_1 \), a material parameter. The boundary conditions associated with the geodesic ow on the two groups we study are given by either¶¶\( u = 0\,{\rm on}\,\partial \Omega \)¶or¶\( u \cdot n = 0\qquad {\rm and}\qquad(\nabla_{n}u)^{\rm tan} + S_{n}(u) = 0\,{\rm on}\,\partial\Omega \),¶where n is the outward pointing unit normal on \( \partial\Omega \), and where Sn is the second fundamental form of \( \partial\Omega \). We prove that for initial data u0 in Hs, 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 \( \langle \cdot,\cdot \rangle_1 \), 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.

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  • J.E. Marsden
    • 1
  • T.S. Ratiu
    • 2
  • S. Shkoller
    • 3
  1. 1.Control and Dynamical Systems, California Institute of Technology, 107-81, Pasadena, CA 91125, USA, e-mail: marsden@cds.caltech.edu<\EAD>US
  2. 2.Département de Mathematiques, Ecole Polytechnique fédérale de Lausanne, CH-1015 Lausanne, Switzerland, e-mail: Tudor.Ratiu@epfl.ch<\EAD>CH
  3. 3.Department of Mathematics, University of California, Davis, CA 95616, USA, e-mail: shkoller@math.ucdavis.edu<\EAD>US