Global regularity and convergence of 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 the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equation, remains Lipschitz for all times, (ii) an initially Holder C^{1,beta}, 0 <= beta < 1, chord arc curve remains in C^{1,beta} for all times, and finally, (iii) an initially Holder C^{n,beta}, n <= 1, 0 < beta < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-alpha and the BR-alpha descriptions of the vortex sheet motion are equivalent.

arXiv comment36 pages, no figures, will appear in CPAM, corrected typos, minor clarifications added

References in corpus (11)

• On admissibility criteria for weak solutions of the Euler equations
• The Camassa-Holm equations as a closure model for turbulent channel flow
• Regularization modeling for large-eddy simulation
• Direct Numerical Simulations of the Navier-Stokes Alpha Model
• 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
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Cited by in corpus (4)

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• Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers
• Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincaré equations
• Mathematics for 2d Interfaces