Axisymmetric Euler-$\alpha$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions

TL;DR

This paper proves the global existence and uniqueness of weak solutions for the axisymmetric Euler-$ alpha$ equations without swirl, extending the understanding of such flows and solutions with Radon measure initial vorticity.

Contribution

It establishes the first known results on global existence and uniqueness for 3D Euler-$ alpha$ equations without swirl with Radon measure initial data.

Findings

Global existence of weak solutions with Radon measure initial vorticity.

Uniqueness of solutions when initial vorticity is in $L^p_c$ for $p > 3/2$.

No prior results available for similar 3D Euler equations.

Abstract

The global existence of weak solutions for the three-dimensional axisymmetric Euler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$ satisfies $\frac{\nabla \times v_0}{r}$ is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided $\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$ with $p>{3/2}$. It is worth mention that no such results are known to be available, so far, for the three-dimensional Euler equations of ideal incompressible flows.