Invariant measures for stochastic damped 2D Euler equations

TL;DR

This paper investigates the stochastic damped 2D Euler equations, establishing the existence of invariant measures in the space of essentially bounded vorticity functions using advanced probabilistic and functional analysis techniques.

Contribution

It proves the existence of invariant measures for stochastic damped 2D Euler equations in $L^ abla$ space, extending previous results on weak solutions and pathwise uniqueness.

Findings

Existence of invariant measures in $L^ abla$ space.

The Markov property for solutions is established.

Application of Krylov-Bogoliubov's method in weak$ abla$ topology.

Abstract

We study the two-dimensional Euler equations, damped by a linear term and driven by an additive noise. The existence of weak solutions has already been studied; pathwise uniqueness is known for solutions that have vorticity in $L^\infty$. In this paper, we prove the Markov property and then the existence of an invariant measure in the space $L^\infty$ by means of a Krylov-Bogoliubov's type method, working with the weak$\star$ and the bounded weak$\star$ topologies in $L^\infty$.