A note on Prandtl boundary layers

Yan Guo, Toan Nguyen

Research output: Contribution to journalArticle

63 Citations (Scopus)

Abstract

This note concerns nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gérard-Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl equation is not well-posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.

Original languageEnglish (US)
Pages (from-to)1416-1438
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number10
DOIs
StatePublished - Oct 1 2011

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Ill-posedness
Boundary Layer
Boundary layers
Sobolev spaces
Shear flow
Shear Flow
Incompressible Flow
Well-posedness
Incompressible Fluid
Monotonic
Sobolev Spaces
Fluid Flow
Flow of fluids
Valid
Verify

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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A note on Prandtl boundary layers. / Guo, Yan; Nguyen, Toan.

In: Communications on Pure and Applied Mathematics, Vol. 64, No. 10, 01.10.2011, p. 1416-1438.

Research output: Contribution to journalArticle

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