, Volume 10, Issue 3, pp 582-599

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

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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) = u 0,¶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 S n is the second fundamental form of \( \partial\Omega \) . 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 \( \langle \cdot,\cdot \rangle_1 \) , 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.

Submitted: November 1998, Revised version: Februar 1999, Final version: July 1999.