Differential geometry on diffeomorphism groups and Lagrangian stability of viscous flows
Let \(\nabla\) be a connection on a manifold \(M\) and \(\tau\) be a \((1,1)\) tensor field on \(M\) (i.e., a field of linear operators in tangent spaces). The author introduces the notion of \(\tau\)-geodesic as a curve \(x(t)\) satisfying the equation \(\nabla{\dot x}\dot x+\tau(\dot x)=0\). Then it is shown that the flow of viscous incompressible fluid on a compact manifold is a \(\tau\)-geodesic on the manifold of volume-preserving diffeomorphisms with \(\tau=-{\frac 1R}\Delta\) where \(R\) is Reynolds number and \(\Delta\) is Laplacian. An analogue of the Jacobi equation for such geodesic is investigated in relation with Lagrangian stability. Reviewer's remark: The presentation of the flow of viscous incompressible fluid as a \(\tau\)-geodesic with above \(\tau\) on the group of diffeomorphisms is in fact a new point of view at the description of viscous incompressible fluid given in the famous paper [\textit{D. Ebin} and \textit{J. Marsden}, ``Groups of diffeomorphisms and the motion of an incompressible fluid'', Ann. Math. (2) 92, 102--163 (1970; Zbl 0211.57401)].
scientific article; zbMATH DE number 5134112
scientific article