Variational formulation for incompressible Euler flow / shape-morphing metric and geodesic

Shape variational formulation for Euler flow has already been considered by the author in (1999a, 2007c). We develop here the control approach considering the convection (or mass transport) as the "state equation" while the speed vector field is the control and we introduce the h-Sobolev curvature which turns to be shape differentiable. The value function defines a new shape metric; we derive existence of geodesic for a p-pseudo metric, verifying the triangle property with a factor 2p-1, for any p > 1. Any geodesic solves the Euler equation for incompressible fluids and, in dimension 3, is not curl free. The classical Euler equation for incompressible fluid (3), coupled with the convection (1) turns to have variational solutions when conditions are imposed on the convected tube ζ while no initial condition has to be imposed on the fluid speed V itself.