| Abstract: |
| It is the aim of this talk to present a geometric method to study various model equations appearing in mathematical hydrodynamics as geodesic flows of right-invariant metrics induced by suitable Fourier multipliers on the Fr\\`{e}chet--Lie group of all diffeomorphisms of the $n$-dimensional torus and the Euclidean $n$-space. This approach covers in particular right-invariant metrics induced by Sobolev norms of fractional order. It is shown that the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential mapping is a smooth local diffeomorphism, provided that the symbol complies with certain mild structural conditions. |
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