Groups of Diffeomorphisms for Manifolds with Boundary and Hydrodynamics

Introduction 1 1. A review of the Hilbert manifold of maps and diffeomorphism groups 5 1.1. Notation 7 2. New diffeomorphism subgroups 8 2.1. Neumann boundary conditions for diffeomorphisms 8 2.2. Mixed boundary conditions for diffeomorphisms 12 2.3. Dirichlet boundary conditions for diffeomorphisms 14 2.4. The group exponential map 14 2.5. A unified approach to differentiable structure on subgroups of Ds 15 2.6. The group Ds([0, 1]) 15 3. Hodge and Stokes decompositions on manifolds with boundary 15 4. A new weak invariant metric on Ds μ and its geodesics 18 5. Smoothness of the Geodesic Spray of 〈·, ·〉 on Ds μ,D 22 6. Weak covariant derivative and curvature operators on Ds μ,D 26 7. Geodesic flow and curvature on Ds([0, 1]) 28 8. Geometric analysis of the viscous problem and its regular limit 30 Appendix A. Euler-Poincaré Reduction 31 Appendix B. Smoothness of differential bundle maps over the identity 32 Acknowledgments 36 References 37