Abstract: We consider weak eulerian evolution of domains through the convection of a measurable set by a nonsmooth vector field V. The transverse variation leads to derivative of functional associated to the evolutiontube and we propose eulerian variational formulation for several classical problems such as incompressible euler flow ( in \cite{chemnitz}, \cite{cambridge} minimal curves...which turn to be governed by a geometrical adjoint state lambda which is backward and is obtained with the use of the so-called tranvserse field Z introduced in \cite{washingtown}. We also re-visit the shape different- ial equation introduced in 1976 () and extend it to the level set approach whose speed vector approach was contained in the free boundary modeling in 1980 (\cite{iowa2}).