Highly rotating viscous compressible fluids in presence of capillarity effects
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- 作者：Francesco Fanelli
- 关键词：Mathematics Subject ClassificationPrimary 35Q35 ; Secondary 35B25 ; 35B40 ; 76U05
- 刊名：Mathematische Annalen
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：366
- 期：3-4
- 页码：981-1033
- 全文大小：825 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-1807
- 卷排序：366
文摘
We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain \(\mathbb {R}^2\times \,]0,1[\,\) and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter \(\varepsilon \rightarrow 0\), we perform the incompressible and high rotation limits simultaneously; moreover, we consider both the constant and vanishing capillarity regimes. In this last case, the limit problem is identified as a 2-D incompressible Navier–Stokes equation in the variables orthogonal to the rotation axis; if the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure, due to the presence of an additional surface tension term. In the vanishing capillarity regime, various rates at which the capillarity coefficient goes to 0 are considered: in general, this produces an anisotropic scaling in the system. The proof of the results is based on suitable applications of the RAGE theorem, combined with microlocal symmetrization arguments.