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Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. (English) Zbl 0961.35101

The author reviews resent progress on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solution of the Navier-Stokes equations.
Let \(v(x,y,t)=(v_1,v_2)\) be the vector of velocity of a fluid and \(p(x,y,t)\) be the pressure. Prandtl’s equations \[ \frac{\partial v_1}{\partial t}+v_1 \frac{\partial v_1}{\partial x_1}+v_2 \frac{\partial v_1}{\partial x_2}-\frac{\partial^2 v_1}{\partial x^2_2}+ \frac{\partial p}{\partial x_1}=0,\quad \frac{\partial v_1}{\partial x_1}+\frac{\partial v_2}{\partial x_2}=0, \] with the initial condition \(v(x,0)=v_0(x)\) are considered in \( \Omega=\{x\in \mathbb{R}^2: x_2\geqslant 0\}\). The corresponding steady problem is considered too. The author presents solvability results for these problems including an unpublished one. In addition, the limit of the solution to the Navier-Stokes equations \[ \frac{\partial v^\varepsilon}{\partial t} +(v^\varepsilon\cdot\nabla)v^\varepsilon -\varepsilon\Delta v^\varepsilon+p^\varepsilon=0, \quad \nabla\cdot v^\varepsilon=0 \quad\text{in} \;\Omega \] for \(\varepsilon\to 0\) is discussed.
Finally, the author indicates some directions where progress is expected in the near future. The proofs are not presented.

MSC:

35Q30 Navier-Stokes equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:

[1] L Prandtl. Verhandlung des III Internationalen Mathematiker-Kongresses (Heidelberg, 1904), p. 484-491
[2] K Nickel. Prandtl’s boundary layer theory from the viewpoint of a mathematician. Ann Rev Fluid Mech, 1973, 5:405-428 · Zbl 0269.76017
[3] H Johnston, J G Liu, Weinan E. The infinite Reynolds number limit of ow past cylinder. in preparation
[4] O A Oleinik. The Prandtl system of equations in boundary layer theory. Soviet Math Dokl, 1963, 4:583-586 · Zbl 0134.45004
[5] S Goldstein. On laminar boundary layer ow near a point of separation. Quart J Mech Appl Math, 1948, 1:43-69 · Zbl 0033.31701
[6] K Stewartson. On Goldstein’s theory of laminar separation. Quart J Mech Appl Math, 1958, 11:399-410 · Zbl 0173.28102
[7] L Caffarelli, Weinan E. Separation of steady boundary layers. unpublished
[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, 8:775-779
[9] Weinan E, B Engquist. Blowup of solutions to the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, L:1287-1293 · Zbl 0908.35099
[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, 463-491 · Zbl 0913.35102
[11] E Grenier. On the instability of boundary layers of Euler equations. In press · Zbl 0958.35106
[12] E Grenier, O Gues. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J Diff Eq, 1998, 143:110-146 · Zbl 0896.35078
[13] D Serre. Systems of conservation laws, II. to be published
[14] Z Xin. Viscous boundary layers and their stability (I). J PDEs, 1998, 11:97-124 · Zbl 0906.35057
[15] R Temam, X M Wang. Boundary layers for the Navier-Stokes equations with non-characteristic boundary. In press · Zbl 0997.35042
[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, 85-98 · Zbl 0559.35067
[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{828 · Zbl 1043.35127
[18] N Masmoudi. About the Navier-Stokes and Euler systems and rotating fluids with boundary. In press · Zbl 0915.76017
[19] Y Brenier. In press
[20] J Hunter. private communication
[21] O A Oleinik, Samokhin. Mathematical Methods in Boundary Layer Theory. Phismathgis”Nauka”, Moscow, 1997
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