Publication 25-CNA-021

Incompressible 2D Euler Equations With Non-Decaying Random Initial Vorticity

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Milton C. Lopes Filho
Instituto de Matemática
Universidade Federal do Rio de Janeiro
Rio de Janeiro, Brazil
mlopes@im.ufrj.br

Helena J. Nussenzveig Lopes
Instituto de Matemática
Universidade Federal do Rio de Janeiro
Rio de Janeiro, Brazil
hlopes@im.ufrj.br

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 $\mathbb{R}^2$ 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.

Get the paper in its entirety as  25-CNA-021.pdf

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