Invariant measures of the 2D Euler and Vlasov equations

TL;DR

This paper investigates invariant measures for the 2D Euler and Vlasov equations, deriving explicit forms and analyzing their properties, stability, and implications for ergodicity in these Hamiltonian systems.

Contribution

It provides a detailed derivation of microcanonical measures as Young measures for 2D Euler equations and extends the analysis to Vlasov equations, including proofs of invariance and uniqueness of equilibria.

Findings

Microcanonical measures correspond to maximization of mean-field entropy.

Explicit proof of invariance of these measures for 2D Euler dynamics.

Extension of results to Vlasov equations with convex potentials.

Abstract

We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.