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

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

2009-07-14

arxiv.org/abs/0907.2348v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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