Abstract
This paper is concerned with the question of convergence of the nonstationary, incompressible Navier-Stokes flow u = u v to the Euler flow u as the viscosity v tends to zero. If the underlying space domain is all of Rm, the convergence has been proved by several authors under appropriate assumptions on the convergence of the data (initial condition and external force); see Golovkin [1] and McGrath [2] for m = 2 and all time, and Swann [3] and the author [4,5] for m = 3 and short time. The case m ⩾ 4 can be handled in the same way; in fact, the simple method given in [5] applies to any dimension. All these results refer to strong solutions (or even classical solutions, depending on the data) of the Navier-Stokes equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. K. Golovkin, Vanishing viscosity in Cauchy’s problem for hydronamics equations, Proc. Steklov Inst. Math. (English translation) 92 (1966), 33–53.
F. J. McGrath, Non-stationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1968), 329–348.
H. Swann, The convergence with vanishing viscosity of non-stationary Navier-Stokes flow to ideal flow in R3, Trans. Amer. Math. Soc. 157 (1971), 373–397.
T. Kato, Non-stationary flows of viscous and ideal fluids in R3, J. Functional Anal. 9 (1972), 296–305.
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes In Math. 448, Springer 1975, 25–70.
J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), 193–248.
E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231.
O. A. Ladyzenskaya, The mathematical theory of viscous incompressible flow, Second English Edition, Gordon and Breach, New York 1969.
J. L. Lions, Quelque méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris 1969.
R. Temam, Navier-Stokes equations, North-Holland Amsterdam-New York-Oxford 1979.
J. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976), 61–102.
D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92 (1970), 102–163.
J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Anal. 15 (1974), 341–363.
R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Anal. 20 (1975), 32–43.
C. Y. Lai, Studies on the Euler and the Navier-Stokes equations, Thesis, University of California, Berkeley, 1975.
T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200.
T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Functional Anal. (1984), to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Science+Business Media New York
About this paper
Cite this paper
Kato, T. (1984). Remarks on Zero Viscosity Limit for Nonstationary Navier- Stokes Flows with Boundary. In: Chern, S.S. (eds) Seminar on Nonlinear Partial Differential Equations. Mathematical Sciences Research Institute Publications, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1110-5_6
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1110-5_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7013-3
Online ISBN: 978-1-4612-1110-5
eBook Packages: Springer Book Archive