Invariant measures of the 2D Euler and Vlasov equations

Abstract

We discuss invariant measures of partial differential equations such as the 2D Euler orVlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid forN-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measurewhere N goes to infinity. When only the energy and enstrophy invariants are taken into account, wegive an explicit computation to prove the following result: the microcanonical measure isactually a Young measure corresponding to the maximization of a mean-field entropy. Weexplain why this result remains true for more general microcanonical measures, when allthe dynamical invariants are taken into account. We give an explicit proof that thesemicrocanonical measures are invariant measures for the dynamics of the 2D Eulerequations. We describe a more general set of invariant measures and discuss briefly theirstability and their consequence for the ergodicity of the 2D Euler equations. The extensionof these results to the Vlasov equations is also discussed, together with a proof of theuniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials.Even if we consider, in this paper, invariant measures only for Hamiltonian equations, withno fluxes of conserved quantities, we think this work is an important step towards thedescription of non-equilibrium invariant measures with fluxes.