##
**Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid.**
*(English)*
Zbl 1044.35061

Let \({\mathcal D}^s\) be the topological group of Hilbert \(H^s\)-class volume preserving diffeomorphisms of a compact, oriented, \(C^\infty\)-smooth, \(n\)-dimensional Riemannian manifold with boundary. The existence of three new subgroups \({\mathcal D}^s_{\mu,D}\), \({\mathcal D}^s_{\mu,N}\), and \({\mathcal D}^s_{\mu,\text{mix}}\) associated with the Dirichlet, Neumann and mixed type boundary conditions that arise in second-order elliptic differential equations is established. Endowed with the \(H^s\)-class topologies for \(s> {n\over 2}+ 1\), they become \(C^\infty\)-differential manifolds with \(H^1\)-equivariant right invariant metric. The unique smooth geodesics \(\eta(t,.)\) are the flows of a velocity vector field \(u(t,x)\) (i.e., \(\partial_t \eta(t,.)= u(t,\eta(t,.))\)), where the vector field \(u(t,x)\) solves the so-called Lagrangian averaged Euler equation
\[
\partial_t(1- \alpha^2\Delta_r)u- \nu\Delta_r u+ \nabla_u(1- \alpha^2\Delta_r)u,
\]

\[ -\alpha^2(\nabla u)^t\cdot\Delta_r u= -\text{grad\,}p, \] where \(\Delta_r= -(d\delta+\delta d)+ 2\text{\,Ric}\). This equation, together with the averaged Navier-Stokes equation (not stated here) model the motion of an ideal incompressible fluid without the use of any artificial viscosity and dissipation.

The article provides essential contribution to the turbulence modelling on Riemann manifolds with boundary. The local well-posedness in any dimension and a global result in dimension two are proved by using remarkable geometrical arguments.

\[ -\alpha^2(\nabla u)^t\cdot\Delta_r u= -\text{grad\,}p, \] where \(\Delta_r= -(d\delta+\delta d)+ 2\text{\,Ric}\). This equation, together with the averaged Navier-Stokes equation (not stated here) model the motion of an ideal incompressible fluid without the use of any artificial viscosity and dissipation.

The article provides essential contribution to the turbulence modelling on Riemann manifolds with boundary. The local well-posedness in any dimension and a global result in dimension two are proved by using remarkable geometrical arguments.

Reviewer: Jan Chrastina (Brno)

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

58J32 | Boundary value problems on manifolds |

37K65 | Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76F20 | Dynamical systems approach to turbulence |