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.
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Submitted: November 1998, Revised version: Februar 1999, Final version: July 1999.
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Marsden, J., Ratiu, T. & Shkoller, S. The geometry and analysis of the averaged Euler equations and a new diffeomorphism group . GAFA, Geom. funct. anal. 10, 582–599 (2000). https://doi.org/10.1007/PL00001631
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DOI: https://doi.org/10.1007/PL00001631