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arXiv 1403.0980math.AP
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带表面张力的自由边界Navier-斯托克斯方程的一致正则性

Uniform Regularity for free-boundary navier-stokes equations with surface tension

Tarek Elgindi, Donghyun Lee

摘要:

本文在$\mathbb{R}^3$中研究了具有表面张力的自由边界Navier-斯托克斯方程的零粘性极限,从而推广了Masmoudi和Schmidt [1]的工作,使之考虑了表面张力。由于边界层的存在,我们不能在通常的索伯列夫空间中达到零粘性极限。实际上,当粘性趋于零时,边界处的法向导数应该爆破。为了解决这个问题,我们解决了所谓的Sobolev余正规空间中的自由边界问题(通过坐标变换固定边界后)。我们证明估计是一致的粘度。在无粘极限处理后,得到了带表面张力的自由边界Euler方程解的局部存在性。这项工作和工作[1]之间的主要区别之一是我们使用时间导数估计和Dirichlet-Neumann算子的某些性质。在即将到来的工作中,我们讨论了我们如何可以采取零粘度和表面张力的同时限制[]。

英文摘要:

We study the zero-viscosity limit of free boundary Navier-Stokes equations with surface tension in $\mathbb{R}^3$ thus extending the work of Masmoudi and Rousset [1] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. One of the main differences between this work and the work [1] is our use of time-derivative estimates and certain properties of the Dirichlet-Neumann operator. In a forthcoming work, we discuss how we can take the simultaneous limit of zero viscosity and surface tension [].

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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