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

2009/07/14 by Jiu, Quansen, Niu, Dongjuan, Titi, Edriss S. +1
#35Q30 #76B47 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.0907.2348

Abstract

The global existence of weak solutions for the three-dimensional axisymmetric Euler-α (also known as Lagrangian-averaged Euler-α) equations, without swirl, is established, whenever the initial unfiltered velocity v₀ satisfies (∇ × v₀)/(r) is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided (∇ × v₀)/(r) ∈ L^p_c(ℝ³) 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.

Related

• Global existence of weak solutions to the 3D incompressible axisymmetric Euler equations without swirl
• Wild solutions of the 3D axisymmetric Euler equations
• Global regularity for some axisymmetric Euler flows in ℝ^d
• Removing Type II singularities off the axis for the 3D axisymmetric Euler equations
• Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in ℝ³