Dynamic Growth Estimates of Maximum Vorticity for 3D Incompressible Euler Equations and the SQG Model

Thomas Y. Hou Caltech Zuoqiang Shi Tsinghua University

Numerical Analysis and Scientific Computing mathscidoc:1709.25017

Discrete and Continuous Dynamical Systems - A, 32, (5), 1449-1463, 2012
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity eld, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fe erman [6, 7] and Deng-Hou-Yu [8, 9].
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@inproceedings{thomas2012dynamic,
  title={Dynamic Growth Estimates of Maximum Vorticity for 3D Incompressible Euler Equations and the SQG Model },
  author={Thomas Y. Hou, and Zuoqiang Shi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170927152609041691836},
  booktitle={Discrete and Continuous Dynamical Systems - A},
  volume={32},
  number={5},
  pages={1449-1463},
  year={2012},
}
Thomas Y. Hou, and Zuoqiang Shi. Dynamic Growth Estimates of Maximum Vorticity for 3D Incompressible Euler Equations and the SQG Model . 2012. Vol. 32. In Discrete and Continuous Dynamical Systems - A. pp.1449-1463. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170927152609041691836.
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