On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

21 Scopus Citations
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Original languageEnglish
Pages (from-to)83-108
Journal / PublicationArchive for Rational Mechanics and Analysis
Volume220
Issue number1
Online published21 Sep 2015
Publication statusPublished - Apr 2016

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.