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

TL;DR

This paper proves the global existence of weak solutions to the 2D Euler equations with a class of non-decaying, randomly initialized vorticities, by constructing suitable initial velocities and leveraging recent well-posedness results.

Contribution

It introduces a novel class of initial vorticities with non-decaying behavior and constructs initial velocities to establish global solutions for almost every such initial condition.

Findings

Global well-posedness for almost every random initial vorticity

Construction of initial velocities with slow growth at infinity

Application of recent well-posedness results to non-decaying vorticities

Abstract

Consider a random initial vorticity $\omega_0(x) = \sum_{n\in \mathbb{Z}^2} a_n \phi(x-n)$, where $\phi$ is bounded and compactly supported and $\{a_n\}$ are independent, uniformly bounded, mean $0$, variance $1$ random variables (i.e. $\omega_0$ 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.