On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid
On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid
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DOI：
10.1007/s00205-020-01516-4
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发表时间：
2019-01
影响因子：
2.5
通讯作者：
M. Disconzi;Chenyun Luo
中科院分区：
文献类型：
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作者：
M. Disconzi;Chenyun Luo
本文建立了具有表面张力的可压缩自由边界Euler方程在液体情况下的不可压缩极限。与最近在Lindblad和Luo（Commun Pure Appl Math 71：1273-1333，2018）和Luo（Ann PDE 4（2）：1-71，2018）中处理的没有表面张力的情况相比，表面张力的存在引入了严峻的新技术挑战，因为当表面张力不存在时自动消失的几个边界项现在在最高阶处起作用。结合的必要性，产生均匀的声速估计，以通过的限制，这样的困难意味着，无论是用于没有表面张力的情况下，也不是估计先前推导出的液体与表面张力和固定的声速，在这里是适用的技术。为了得到我们的结果，我们设计了一个合适的声速加权能量，考虑到耦合的流体运动与边界几何形状。估计关闭充分利用非线性结构的欧拉方程和调用几个几何性质的边界，以产生一些显着的取消。我们强调，我们不假定流体是无旋的。
In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently in Lindblad and Luo (Commun Pure Appl Math 71:1273–1333, 2018) and Luo (Ann PDE 4(2):1–71, 2018), the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surface tension, nor estimates previously derived for a liquid with surface tension and fixed sound speed, are applicable here. In order to obtain our result, we devise a suitable sound-speed-weighted energy that takes into account the coupling of the fluid motion with the boundary geometry. Estimates are closed by exploiting the full non-linear structure of the Euler equations and invoking several geometric properties of the boundary in order to produce some remarkable cancellations. We stress that we do not assume the fluid to be irrotational.