Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Abstract

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.

7 pages, no figures, submitted to the proceedings of the Conference EE250, corrected typos, references added

References in corpus (10)

• The Camassa-Holm equations as a closure model for turbulent channel flow
• The Navier-Stokes-alpha model of fluid turbulence
• A connection between the Camassa-Holm equations and turbulent flows in channels and pipes
• Regularization modeling for large-eddy simulation
• Euler Equations of Incompressible Ideal Fluids
• Analytical Study of Certain Magnetohydrodynamic-alpha Models
• An Eulerian-Lagrangian approach to the Navier-Stokes equations
• On the Clark-alpha model of turbulence: global regularity and long--time dynamics
• Variational Principles for Lagrangian Averaged Fluid Dynamics
• Recent progress in mathematical analysis of vortex sheets

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• Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers
• Adaptive solution of initial value problems by a dynamical Galerkin scheme
• The Navier-Stokes-Voight Model for Image Inpainting
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Axisymmetric Euler-$\alpha$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions