Finite time singularity formation for moving interface Euler equations
Finite time singularity formation for moving interface Euler equations
复制标题
移动界面欧拉方程的有限时间奇点形成
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
D. Coutand
中科院分区：
文献类型：
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作者：
D. Coutand
本文提出了一种新的一般方法，用于求解平面内不可压Euler方程的运动界面问题的有限时间奇异性。
第一个问题考虑的是两相欧拉涡面问题的表面张力，证明了有限时间奇异性的问题的自然范数适当的初始数据。这与[4]中介绍的单相欧拉方程的有限时间飞溅和飞溅奇点形成的情况形成鲜明对比，并在[8]中在更一般的背景下进行了研究，其中自然范数（在单相流体中）一直保持有限直到接触。
考虑的第二个问题涉及存在一个较重的刚体在理想流体中运动。我们首先确定刚体将在有限时间内撞击流体域的底部（在更一般的背景下，并且与[19]中首次解决该问题的方法非常不同）。接下来，我们建立了表面能爆炸，并根据接触区的性质来描述接触时刚体的加速度：加速度与接触时的运动相反，如果接触区包含曲线，则加速度是正有限的，否则是无限的。
This paper proposes a new general methodology for finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane.
The first problem considered is the two-phase Euler vortex sheets problem with surface tension for which is proved the finite time singularity of the natural norm of the problem for suitable initial data. This is in striking contrast with the case of finite time splash and splat singularity formation for the one phase Euler equations introduced in [4] and studied in a more general context in [8], for which the natural norm (in the one phase fluid) stays finite all the way until contact.
The second problem considered involves the presence of a heavier rigid body moving in the perfect fluid. We first establish that the rigid body will hit the bottom of the fluid domain in finite time (in a more general context and very different methodology than first done for this problem in [19]). We next establish that a surface energy blows up, and characterize the acceleration of the rigid body at contact, depending on the nature of the contact zone: Acceleration opposes the motion at contact, and is either positive finite if the contact zone contains a curve, or infinite otherwise.