Well-posedness of vortex sheets with surface tension
Well-posedness of vortex sheets with surface tension
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DOI：
10.1137/s0036141002403869
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发表时间：
2003-01-01
影响因子：
2
通讯作者：
Ambrose, DM
中科院分区：
文献类型：
--
作者：
Ambrose, DM
研究了具有表面张力的二维周期性涡片的初值问题。我们允许上层和下层流体有不同的密度。没有表面张力，旋涡片是病态的：它表现出众所周知的开尔文-亥姆霍兹不稳定性。在线性化的运动方程中，表面张力消除了不稳定性。据推测，表面张力也使整个问题得到适定。我们用能量法证明了这个猜想是正确的。特别是对于具有足够光滑数据的具有表面张力的涡片初值问题，证明了其解在时间上局部存在、唯一且连续依赖于初值数据。该分析使用了侯、洛温格鲁和雪莱的数值研究中的两个重要思想。首先，涡旋片的切角和弧长被使用，而不是笛卡尔变量。其次，为了简化演化方程，使用了一个特殊的切向速度，而不是纯粹的拉格朗日公式。结果的一个特例是具有表面张力的水波的适定性；这是第一个证据（表面张力），允许波浪翻转。
We study the initial value problem for two-dimensional, periodic vortex sheets with surface tension. We allow the upper and lower fluids to have different densities. Without surface tension, the vortex sheet is ill-posed: it exhibits the well-known Kelvin-Helmholtz instability. In the linearized equations of motion, surface tension removes the instability. It has been conjectured that surface tension also makes the full problem well-posed. We prove that this conjecture is correct using energy methods. In particular, for the initial value problem for vortex sheets with surface tension with sufficiently smooth data, it is proved that solutions exist locally in time, are unique, and depend continuously on the initial data. The analysis uses two important ideas from the numerical work of Hou, Lowengrub, and Shelley. First, the tangent angle and arclength of the vortex sheet are used rather than Cartesian variables. Second, instead of a purely Lagrangian formulation, a special tangential velocity is used in order to simplify the evolution equations. A special case of the result is well-posedness of water waves with surface tension; this is the first proof (with surface tension) which allows the wave to overturn.