ISSN:
1420-8970
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00001631
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