Invariant or quasi-invariant probability measures for infinite dimensional groups. I: Non-ergodicity of Euler hydrodynamic (Q1043871)

The paper considers the problem of invariants for flows. Since the motion of an incompressible fluid conserves the volume, a solution of the Euler equation generates a local group of unitary operation on \(L^2\). It is approached the lack of compactness is a bounded domain within the paradigm of Arnold which considers Euler equation as the equation of geodesics on the volume preserving diffeomeorphisms groups; this paradigm is developed on the \(d\)-dimensional torus to imply Fourier series. Finally it is proved that the deterministic Euler flow cannot leave invariant any probability measure.