---

Jean Paul Zolesio (CNRS, France)

Title: Variational formulation for incompressible Euler flow in any dimension

Abstract: We introduce an Energy term E(T) where T is a "tube", a pair T=(h,V), where h is a characteristic function, h(h-1)=0, while V (which is free divergence) is a square integrable vector field ( over a bounded domain ) such that h is convected by V. The only imposed constraints are on h (and none on V). We develop the very simple functional analysis framework in which this energy term reaches its infimum. At such point the vector field V solves the usual incompressible Euler equation for fluid dynamic. By an obvious way we can extend that energy term for a new viscous (Navier-Stokes like) model for incompressible fluid which turn to be variational and physically much more accurate than the Old N.S. modelling. We claim here that the "non variational aspect" of the Navier Stokes equations derives from a poor modelling based on the fact that these equations result in additing two terms, one expressed in eulerian coordinate, the second in Lagrangian coordinate (the laplace operator acting for the viscosity force). In the present modelling, the total energy is expressed in Eulerian viewpoint, i.e. on the speed V. We hope that such a variational approach for fluid dynamic should help for the study of stability of coupled fluid-structures devices, in particular the situation of interest when the geometry of the structure (for example moving wings) is dynamical. In this situation no egeinvalue argument is available as we are faced to a non cylindrical problem (with non autonomous operators).