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Abstract
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 equations with a special structure.
Original language | English |
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Pages (from-to) | 83-108 |
Journal | Archive for Rational Mechanics and Analysis |
Volume | 220 |
Issue number | 1 |
Online published | 21 Sept 2015 |
DOIs | |
Publication status | Published - Apr 2016 |
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Dive into the research topics of 'On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space'. Together they form a unique fingerprint.Projects
- 1 Finished
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GRF: Mathematical Theories on the Prandtl System in Sobolev Spaces
YANG, T. (Principal Investigator / Project Coordinator)
1/01/14 → 6/12/17
Project: Research