On the ill-posedness of the Prandtl equations in three-dimensional space

Cheng-Jie Liu Ya-Guang Wang Tong Yang

Analysis of PDEs mathscidoc:1912.43959

Archive for Rational Mechanics and Analysis, 220, (1), 83-108, 2016.4
In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308 , 2014) on the well-posedness theory for the three-dimensional Prandtl
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@inproceedings{cheng-jie2016on,
  title={On the ill-posedness of the Prandtl equations in three-dimensional space},
  author={Cheng-Jie Liu, Ya-Guang Wang, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210918115888523},
  booktitle={Archive for Rational Mechanics and Analysis},
  volume={220},
  number={1},
  pages={83-108},
  year={2016},
}
Cheng-Jie Liu, Ya-Guang Wang, and Tong Yang. On the ill-posedness of the Prandtl equations in three-dimensional space. 2016. Vol. 220. In Archive for Rational Mechanics and Analysis. pp.83-108. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210918115888523.
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