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On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity. (English) Zbl 1043.35127

Summary: We establish partial results concerning the convergence of the solutions of the Navier-Stokes equations to that of the Euler equations. Namely, we prove convergence on any finite interval of time, in space dimension two, under a physically reasonable assumption. We consider the flow in a channel or the flow in a general bounded domain.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] S.N. Alekseenko , Existence and asymptotic representation of weak solutions to the flowing problem under the condition of regular slippage on solid walls , Siberian Math. J. 35 ( 1994 ), 209 - 229 . MR 1288259 | Zbl 0856.35099 · Zbl 0856.35099
[2] K. Asano , Zero-Viscosity Limit of the Incompressible Navier-Stokes Equations 1, 2 , Preprint, University of Kyoto , 1997 .
[3] R. Balian - J.L. Peube , ” Fluid Dynamics ”, Cours de l’Ecole d’Ete de Physique Théorique , Les Houches Gordon and Breach Science Publishers , New-York , 1977 . MR 495783 | Zbl 0348.00025 · Zbl 0348.00025
[4] G.I. Barenblatt - A.J. Chorin , Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow , Proc. Natl. Acad. Sci., Applied Mathematics 93 ( 1996 ). Zbl 0856.76026 · Zbl 0856.76026
[5] G.I. Barenblatt - A.J. Chorin , Scaling laws and zero viscosity limits for wall-bounded shear flows and for local structure in developed turbulence . Preprint, Center for Pure and Applied Mathematics, University of California at Berkeley , n^\circ PAM-678, 1996 . MR 1492690 · Zbl 0866.76035
[6] G.K. Batchelor , ” An Introduction to Fluid Dynamics ”, Cambridge University Press , Cambridge , 1967 . MR 1744638 | Zbl 0152.44402 · Zbl 0152.44402
[7] J.T. Beale - T. Kato - A. Majda , Remarks on the breakdown of smooth solutions for the 3D Euler equations , Comm. Math. Phys. 94 ( 1984 ), 61 - 66 . Article | MR 763762 | Zbl 0573.76029 · Zbl 0573.76029
[8] A.J. Chorin , Turbulence as a near-equilibrium process , in Lecture in Applied Mathematics 31 ( 1996 ), 235 - 249 , AMS , Providence . MR 1363031 | Zbl 0838.76034 · Zbl 0838.76034
[9] P. Constantin - J. Wu , Inviscid limit for vortex patches , Nonlinearity 8 ( 1995 ), 735 - 742 . MR 1355040 | Zbl 0832.76011 · Zbl 0832.76011
[10] Weinan E. - B. Engquist , Blow-up of Solutions of the Unsteady Prandtl’s Equation , preprint, 1996 .
[11] D.G. Ebin - J. Marsden , Groups of diffeomorphisms and the motion of an incompressible fluid , Arch. Rational Mech. Anal. 46 ( 1972 ), 241 - 279 . MR 426034 · Zbl 0211.57401
[12] W. Eckhaus , ” Asymptotic Analysis of Singular Perturbations ”, North-Holland , 1979 . MR 553107 | Zbl 0421.34057 · Zbl 0421.34057
[13] P. Fife , Considerations regarding the mathematical basis for Prandtl’s boundary layer theory , Arch. Rational Mech. Anal. 38 ( 1967 ), 184 - 216 . MR 227633 | Zbl 0172.53801 · Zbl 0172.53801
[14] P. Germain , ” Méthodes Asymptotiques en Mécanique des Fluides ”, in [BP]. Zbl 0387.76001 · Zbl 0387.76001
[15] H.P. Greenspan , ” The Theory of Rotating Fluids ”, Cambridge Univ. Press , Cambridge , 1968 . MR 639897 | Zbl 0182.28103 · Zbl 0182.28103
[16] K. Kato , ” Remarks on the Zero Viscosity Limit for Nonstationary Navier-Stokes Flows with Boundary ”, in Seminar on PDE, edited by S. S. Chern, Springer , N.Y. , 1984 . MR 765230 · Zbl 0559.35067
[17] T. Kato , On the classical solutions of two dimensional nonstationary Euler equations , Arch. Rational Mech. Anal 25 ( 1967 ), 188 - 200 . MR 211057 | Zbl 0166.45302 · Zbl 0166.45302
[18] P. Lagerström , ” Matched Asymptotics Expansion, Ideas and Techniques ”, Springer-Verlag , New-York , 1988 . MR 958913 | Zbl 0666.34064 · Zbl 0666.34064
[19] L. Landau - E. Lifschitz , ” Fluid Mechanics ”, Addison-Wesley , New-York , 1953 .
[20] L. Lichtenstein , ” Grundlagen der Hydromechanik ”, Springer-Verlag , 1923 . MR 228225 | Zbl 0157.56701 | JFM 55.1124.01 · Zbl 0157.56701
[21] J.L. Lions , ” Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires ”, Dunod , Paris , 1969 . MR 259693 | Zbl 0189.40603 · Zbl 0189.40603
[22] J.L. Lions , Perturbations singulières dans les problèmes aux limites et en contrôle optimal , Lecture Notes in Math. 323 Springer-Verlag , New-York , 1973 . MR 600331 | Zbl 0268.49001 · Zbl 0268.49001
[23] A. Majda , ”Compressible fluid flow and systems of conservation laws in several space variables” , Springer-Verlag , New-York , 1984 . MR 748308 | Zbl 0537.76001 · Zbl 0537.76001
[24] O. Oleinik , The Prandtl system of equations in boundary layer theory , Dokl. Akad. Nauk S.S.S.R. 150 , 4 ( 3 ) ( 1963 ), 583 - 586 . Zbl 0134.45004 · Zbl 0134.45004
[25] L. Prandtl , Veber Flüssigkeiten bei sehr kleiner Reibung , Verh. III Intern. Math. Kongr. Heidelberg ( 1905 ), 484 - 491 , Teuber , Leibzig . JFM 36.0800.02 · JFM 36.0800.02
[26] M. Sammartino - R.E. Caflish , Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space , I and II Preprint, 1996 . MR 1461106
[27] R. Temam , On the Euler equations of incompressible perfect fluids , J. Funct. Anal. 20 ( 1975 ), 32 - 43 ; Remarks on the Euler equations , in Proc. of ”Symposia in Pure Mathematics”, F. Browder ed., AMS , Providence , vol. 45 ( 1986 ), 429 - 430 . · Zbl 0309.35061
[28] R. Temam , Local existence of C\infty solutions of the Euler equations of incompressible perfect fluids , in Proc. Conf. on ”Turbulence and Navier-Stokes”, Lecture Notes in Mathematics 565 Springer-Verlag , 1976 . Zbl 0355.76017 · Zbl 0355.76017
[29] R. Temam , Behaviour at time t = 0 of the solutions of semi-linear evolution equations , J. Diff. Equ. 17 ( 1982 ), 73 - 92 . MR 645638 | Zbl 0446.35057 · Zbl 0446.35057
[30] R. Temam , Navier-Stokes Equations and Nonlinear Functional Analysis , CBMS-NSF Regional Conference Series in ”Applied Mathematics”, SIAM , Philadelphia . Second edition, 1995 . MR 1318914 | Zbl 0833.35110 · Zbl 0833.35110
[31] R. Temam - X. Wang , Asymptotic analysis of the linearized Navier-Stokes equations in a channel , Differential and Integral Equations , 8 ( 1995 ), 1591 - 1618 . MR 1347972 | Zbl 0832.35112 · Zbl 0832.35112
[32] R. Temam - X. Wang , Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain , Asymptotic Analysis 9 ( 1996 ), 1 - 30 . MR 1383673 | Zbl 0889.35076 · Zbl 0889.35076
[33] R. Temam - X. Wang , Asymptotic analysis of Oseen type equations in a channel at small viscosity , Indiana Univ. Math. J. 45 ( 1996 ), 863 - 916 . MR 1422110 | Zbl 0881.35097 · Zbl 0881.35097
[34] R. Temam - X. Wang , Boundary layers for Oseen’s type equation in space dimension three, dedicated to M. Vishik , Russian Journal of Mathematical Physics , to appear. Zbl 0912.35125 · Zbl 0912.35125
[35] R. Temam - X. Wang , The convergence of the solutions of the Navier-Stokes equations to that of the Euler equations , Applied. Math. Letters , to appear. MR 1471313 | Zbl 0888.35077 · Zbl 0888.35077
[36] M. Van Dyke , ” Perturbation Methods in Fluid Mechanics ”, Academic Press , New-York . MR 416240 | Zbl 0136.45001 · Zbl 0136.45001
[37] M.I. Vishik - L.A. Lyusternik , Regular degeneration and boundary layer for linear differential equations with small parameter , Uspekki Mat. Nauk 12 ( 1957 ), 3 - 122 . MR 96041 | Zbl 0087.29602 · Zbl 0087.29602
[38] W. Wolibner , Un théorème sur l’existence du mouvement plan d’un fluide parfait homogène incompressible, pendant un temps infiniment long , Math. Z. 39 ( 1933 ), 698 - 726 . MR 1545430 | Zbl 0008.06901 | JFM 59.1447.02 · Zbl 0008.06901
[39] T. Yanagisawa - Z. Xin , Singular perturbation of Euler equations with characteristic boundary condition , in preparation.
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