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Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations
Author(s) -
Bardos Claude,
Titi Edriss S.,
Linshiz Jasmine S.
Publication year - 2010
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20305
Subject(s) - mathematics , vorticity , euler equations , mathematical analysis , vortex , lipschitz continuity , euler's formula , vortex sheet , burgers vortex , physics , mechanics
We present an α‐regularization of the Birkhoff‐Rott equation (BR‐α equation), induced by the two‐dimensional Euler‐α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler‐α 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‐α equation, remains Lipschitz for all times, (ii) an initially Hölder C 1, β , 0 ≤ β < 1, chord arc curve remains in C 1, β for all times, and finally, (iii) an initially Hölder C n , β , n ≥ 1, 0 < β < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler‐α and the BR‐α descriptions of the vortex sheet motion are equivalent. © 2009 Wiley Periodicals, Inc.