On a non-periodic modified Euler equation: existence and quasi-invariant measures

TL;DR

This paper proves the existence of global solutions for a modified Euler equation in two dimensions and constructs quasi-invariant measures, enhancing understanding of flow behavior for less regular initial conditions.

Contribution

It establishes global existence of solutions and constructs Gibbs-type measures that are quasi-invariant under the flow, extending analysis to less regular initial data.

Findings

Existence of weak global solutions for bounded initial conditions.

Construction of Gibbs-type measures quasi-invariant under the flow.

Global flow defined almost everywhere with respect to these measures.

Abstract

We consider a modified Euler equation on $\mathbb R^2$. We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be also globally defined.