Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations
Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations
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二维欧拉方程涡片动力学的 Birkhoff-Rott-α 近似的全局正则性和收敛性
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Jasmine Linshiz
中科院分区：
文献类型：
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作者：
C. Bardos;E. Titi;Jasmine Linshiz
本文给出了由二维Euler-α方程导出的Birkhoff-Rott方程（BR-α方程）的α正则化方法，用于涡面动力学。我们证明了Euler-α方程的解收敛到Euler方程的弱解，其中初始涡量是固定符号的有限Radon测度，包括涡面的情况。我们还证明，如果初始涡度密度是关于弧长测度的曲线上的可积函数，（i）在BR-α方程下演化的初始Lipschitz弦弧涡面（曲线）始终保持Lipschitz，（ii）初始Hölder C1，β，0 ≤ β < 1，弦弧曲线始终保持C1，β，最后，（iii）初始Hölder Cn，β，n ≥ 1，0 < β < 1，闭合弦弧曲线始终保持如此。在所有这些情况下，涡面运动的弱欧拉-α和BR-α描述是等价的。© 2009威利期刊公司.
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 C1, β, 0 ≤ β < 1, chord arc curve remains in C1, β for all times, and finally, (iii) an initially Hölder Cn, 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.
影响因子：
2.5
作者：
De Lellis, Camillo;Szekelyhidi, Laszlo, Jr.
通讯作者：
Szekelyhidi, Laszlo, Jr.