Singularity formation for a fluid mechanics model with nonlocal velocity
Singularity formation for a fluid mechanics model with nonlocal velocity
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DOI：
10.4310/cms.2019.v17.n7.a2
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发表时间：
2017-08
影响因子：
1
通讯作者：
Changhui Tan
中科院分区：
文献类型：
--
作者：
Changhui Tan
研究了一维非局部速度流体力学模型。该方程可以看作是一个分数阶多孔介质流，是准地转方程的一维模型，也是欧拉-对准系统的一个特例。对于严格正的光滑初值，Do，Kiselev，Ryzhik和Tan证明了全局正则性.我们构造了一族非负光滑的初始数据，使得解失去了C^1 $正则性。结果表明，严格正性是保证系统全局正则性的一个关键条件。我们还将我们的构造扩展到相应的多维模型。
We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly positive smooth initial data, global regularity has been proved by Do, Kiselev, Ryzhik and Tan. We construct a family of non-negative smooth initial data so that solution loses $C^1$ regularity. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.