Acta Mathematica Sinica

, Volume 16, Issue 2, pp 207–218 | Cite as

Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation

  • E WeinanEmail author


A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero. This is particularly important when boundaries are present since vorticity is typically generated at the boundary as a result of boundary layer separation. The boundary layer theory, developed by Prandtl about a hundred years ago, has become a standard tool in addressing these questions. Yet at the mathematical level, there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory. In this article, we review recent progresses on the analysis of Prandtl's equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation. We also discuss some directions where progress is expected in the near future.


Boundary layer Blow-up Zero-viscosity limit Prandtl’s equation 

1991 MR Subject Classification

35B05 76D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L Prandtl. Verhandlung des III Internationalen Mathematiker-Kongresses (Heidelberg, 1904), p. 484-491Google Scholar
  2. 2.
    K Nickel. Prandtl’s boundary layer theory from the viewpoint of a mathematician. Ann Rev Fluid Mech, 1973, 5:405-428CrossRefGoogle Scholar
  3. 3.
    H Johnston, J G Liu, Weinan E. The infinite Reynolds number limit of ow past cylinder. in preparationGoogle Scholar
  4. 4.
    O A Oleinik. The Prandtl system of equations in boundary layer theory. Soviet Math Dokl, 1963, 4:583-586Google Scholar
  5. 5.
    S Goldstein. On laminar boundary layer ow near a point of separation. Quart J Mech Appl Math, 1948, 1:43-69zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    K Stewartson. On Goldstein’s theory of laminar separation. Quart J Mech Appl Math, 1958, 11:399-410zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    L Caffarelli, Weinan E. Separation of steady boundary layers. unpublishedGoogle Scholar
  8. 8.
    O A Oleinik. Construction of the solutions of a system of boundary layer equations by the method of straight lines. Soviet Math Dokl, 1967, 8:775-779Google Scholar
  9. 9.
    Weinan E, B Engquist. Blowup of solutions to the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, L:1287-1293Google Scholar
  10. 10.
    M Sammartino, R E Caflisch. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I Comm Math Phys, 192, 433-461, II, Comm Math Phys, 1998, 192, 463-491Google Scholar
  11. 11.
    E Grenier. On the instability of boundary layers of Euler equations. In pressGoogle Scholar
  12. 12.
    E Grenier, O Gues. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J Diff Eq, 1998, 143:110-146zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D Serre. Systems of conservation laws, II. to be publishedGoogle Scholar
  14. 14.
    Z Xin. Viscous boundary layers and their stability (I). J PDEs, 1998, 11:97-124zbMATHMathSciNetGoogle Scholar
  15. 15.
    R Temam, X M Wang. Boundary layers for the Navier-Stokes equations with non-characteristic boundary. In pressGoogle Scholar
  16. 16.
    T Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In: Seminar on Partial Differential Equations, S S Chern eds 1984, 85-98Google Scholar
  17. 17.
    R Temam, X M Wang. On the behavior of the solutions of the Navier-Stokes equations. Annali della Normale Superiore di Pisa, Serie IV, 1998, XXV:807{828Google Scholar
  18. 18.
    N Masmoudi. About the Navier-Stokes and Euler systems and rotating fluids with boundary. In pressGoogle Scholar
  19. 19.
    Y Brenier. In pressGoogle Scholar
  20. 20.
    J Hunter. private communicationGoogle Scholar
  21. 21.
    O A Oleinik, Samokhin. Mathematical Methods in Boundary Layer Theory. Phismathgis“Nauka”, Moscow, 1997Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Department of Mathematics and Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations