Invariant measures for the non-periodic two-dimensional Euler equation

TL;DR

This paper constructs Gaussian invariant measures for the 2D Euler equation on the plane, demonstrating existence, uniqueness, and flow continuity for solutions with initial conditions in a specific local Sobolev space.

Contribution

It introduces a new class of invariant measures for the non-periodic 2D Euler equation and proves key properties like existence, uniqueness, and flow continuity.

Findings

Existence of solutions with initial data in $H^eta_{loc}( ^2)$ for $eta<-1$

Construction of Gaussian invariant measures for the 2D Euler equation

Proof of uniqueness and flow continuity for solutions starting in the measure's support

Abstract

We construct Gaussian invariant measures for the two-dimensional Euler equation on the plane. We show the existence of solution with initial conditions in the support of the measures, namely $H^\beta_{loc}(\R^2)$ with $\beta<-1$. Uniqueness and continuity of the velocity flow are proved.