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.

中文翻译：

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液体情况下具有表面张力的可压缩自由边界欧拉方程的不可压缩极限

在本文中，我们建立了在液体情况下具有表面张力的可压缩自由边界欧拉方程的不可压缩极限。与最近在 Lindblad and 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重的新的技术挑战，因为在没有表面张力时自动消失的几个边界项现在贡献于最高阶。结合产生声速均匀估计以达到极限的必要性，这些困难意味着既没有用于没有表面张力的情况的技术，也没有先前为具有表面张力和固定声速的液体推导出的估计，适用于此处。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，将流体运动与边界几何形状的耦合考虑在内。通过利用欧拉方程的完整非线性结构并调用边界的几个几何特性以产生一些显着的抵消来关闭估计。我们强调我们不假设流体是无旋的。