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

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 $\frac{\nabla \times v_0}{r}\in L_c^p({\mathbb{R}}^3)$ 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.

References in corpus (2)

• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Global Existence and Uniqueness of Weak Solutions of 3-D Euler Equations with Helical Symmetry in the Absence of Vorticity Stretching