Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L Prandtl. Verhandlung des III Internationalen Mathematiker-Kongresses (Heidelberg, 1904), p. 484-491
K Nickel. Prandtl’s boundary layer theory from the viewpoint of a mathematician. Ann Rev Fluid Mech, 1973, 5:405-428
H Johnston, J G Liu, Weinan E. The infinite Reynolds number limit of ow past cylinder. in preparation
O A Oleinik. The Prandtl system of equations in boundary layer theory. Soviet Math Dokl, 1963, 4:583-586
S Goldstein. On laminar boundary layer ow near a point of separation. Quart J Mech Appl Math, 1948, 1:43-69
K Stewartson. On Goldstein’s theory of laminar separation. Quart J Mech Appl Math, 1958, 11:399-410
L Caffarelli, Weinan E. Separation of steady boundary layers. unpublished
O A Oleinik. Construction of the solutions of a system of boundary layer equations by the method of straight lines. Soviet Math Dokl, 1967, 8:775-779
Weinan E, B Engquist. Blowup of solutions to the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, L:1287-1293
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, 463-491
E Grenier. On the instability of boundary layers of Euler equations. In press
E Grenier, O Gues. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J Diff Eq, 1998, 143:110-146
D Serre. Systems of conservation laws, II. to be published
Z Xin. Viscous boundary layers and their stability (I). J PDEs, 1998, 11:97-124
R Temam, X M Wang. Boundary layers for the Navier-Stokes equations with non-characteristic boundary. In press
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, 85-98
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{828
N Masmoudi. About the Navier-Stokes and Euler systems and rotating fluids with boundary. In press
Y Brenier. In press
J Hunter. private communication
O A Oleinik, Samokhin. Mathematical Methods in Boundary Layer Theory. Phismathgis“Nauka”, Moscow, 1997
Author information
Authors and Affiliations
Corresponding author
Additional information
Also at Courant Institute, New York University
Rights and permissions
About this article
Cite this article
Weinan, E. Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation. Acta Math Sinica 16, 207–218 (2000). https://doi.org/10.1007/s101140000034
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s101140000034