On a Non-Periodic Modified Euler Equation: Well-Posedness and Quasi-Invariant 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 globally defined.