Mathematics – Analysis of PDEs
Scientific paper
2009-02-19
Mathematics
Analysis of PDEs
36 pages, no figures, will appear in CPAM, corrected typos, minor clarifications added
Scientific paper
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.
Bardos Claude
Linshiz Jasmine S.
Titi Edriss S.
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