Incompressible 2D Euler equations with non-decaying random initial vorticity

Abstract

Consider a random initial vorticity $\ddot{I}{‰}_0(x)={\sum }_{n\in {\mathbb{Z}}^2}{a}_n\ddot{I}•(x-n)$, where $\ddot{I}•$ is bounded and compactly supported and $\{a_n\}$ are independent, uniformly bounded, mean $0$, variance $1$ random variables (i.e. $\ddot{I}{‰}_0$ is an array of randomly weighted vortex blobs). We prove global well-posedness of weak solutions to the Euler equations in ${\mathbf{R}}^2$ for almost every such initial vorticity. The main contribution of our work is the construction of a corresponding initial velocity field that grows slowly at infinity, which enables us to apply a recent well-posedness result of Cobb and Koch.

17 pages, 0 figures