Viscous splitting for the unbounded problem of the NavierStokes equations
Author:
LungAn Ying
Journal:
Math. Comp. 55 (1990), 89113
MSC:
Primary 35Q30; Secondary 65N99, 76D05, 76D07
MathSciNet review:
1023053
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The viscous splitting for the exterior initialboundary value problems of the NavierStokes equations is considered. It is proved that the approximate solutions are uniformly bounded in the space , , and converge with a rate of in the space , where k is the length of the time steps.
 [1]
Robert
A. Adams, Sobolev spaces, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. Pure and
Applied Mathematics, Vol. 65. MR 0450957
(56 #9247)
 [2]
S.
Agmon, A.
Douglis, and L.
Nirenberg, Estimates near the boundary for solutions of elliptic
partial differential equations satisfying general boundary conditions.
I, Comm. Pure Appl. Math. 12 (1959), 623–727.
MR
0125307 (23 #A2610)
 [3]
Giovanni
Alessandrini, Avron
Douglis, and Eugene
Fabes, An approximate layering method for the NavierStokes
equations in bounded cylinders, Ann. Mat. Pura Appl. (4)
135 (1983), 329–347 (1984) (English, with Italian
summary). MR
750540 (86a:35007), http://dx.doi.org/10.1007/BF01781075
 [4]
J.
Thomas Beale and Andrew
Majda, Rates of convergence for viscous
splitting of the NavierStokes equations, Math.
Comp. 37 (1981), no. 156, 243–259. MR 628693
(82i:65056), http://dx.doi.org/10.1090/S00255718198106286930
 [5]
G.
Benfatto and M.
Pulvirenti, Convergence of ChorinMarsden product formula in the
halfplane, Comm. Math. Phys. 106 (1986), no. 3,
427–458. MR
859819 (88a:35186)
 [6]
Alexandre
Joel Chorin, Numerical study of slightly viscous flow, J.
Fluid Mech. 57 (1973), no. 4, 785–796. MR 0395483
(52 #16280)
 [7]
Hiroshi
Fujita and Tosio
Kato, On the NavierStokes initial value problem. I, Arch.
Rational Mech. Anal. 16 (1964), 269–315. MR 0166499
(29 #3774)
 [8]
Tosio
Kato, On classical solutions of the twodimensional nonstationary
Euler equation, Arch. Rational Mech. Anal. 25 (1967),
188–200. MR 0211057
(35 #1939)
 [9]
O.
A. Ladyzhenskaya, The mathematical theory of viscous incompressible
flow, Second English edition, revised and enlarged. Translated from
the Russian by Richard A. Silverman and John Chu. Mathematics and its
Applications, Vol. 2, Gordon and Breach, Science Publishers, New
YorkLondonParis, 1969. MR 0254401
(40 #7610)
 [10]
J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, SpringerVerlag, 1972.
 [11]
F.
J. McGrath, Nonstationary plane flow of viscous and ideal
fluids, Arch. Rational Mech. Anal. 27 (1967),
329–348. MR 0221818
(36 #4870)
 [12]
Roger
Temam, On the Euler equations of incompressible perfect
fluids, J. Functional Analysis 20 (1975), no. 1,
32–43. MR
0430568 (55 #3573)
 [13]
Long
An Ying, Viscosity splitting method in bounded domains, Sci.
China Ser. A 32 (1989), no. 8, 908–921. MR 1055308
(91h:65201)
 [14]
L.a. Ying, The viscosity splitting method for the NavierStokes equations in bounded domains, Science Report, Department of Mathematics and Institute of Mathematics, Peking University, October 1986.
 [15]
Lung
An Ying, On the viscosity splitting method for initialboundary
value problems of the NavierStokes equations, Chinese Ann. Math. Ser.
B 10 (1989), no. 4, 487–512. A Chinese summary
appears in Chinese Ann.\ Math.\ Ser.\ A 10 (1989), no.\ 5, 641. MR 1038384
(91c:35123)
 [16]
Lungan
Ying, Viscosity splitting method for threedimensional
NavierStokes equations, Acta Math. Sinica (N.S.) 4
(1988), no. 3, 210–226. A Chinese summary appears in Acta Math.
Sinica 32 (1989), no. 4, 575. MR 965569
(90b:65193), http://dx.doi.org/10.1007/BF02560577
 [17]
Lungan
Ying, Convergence study for viscous splitting in bounded
domains, Numerical methods for partial differential equations
(Shanghai, 1987), Lecture Notes in Math., vol. 1297, Springer,
Berlin, 1987, pp. 184–202. MR 929048
(89h:65163), http://dx.doi.org/10.1007/BFb0078550
 [1]
 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
 [2]
 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 17 (1959), 623727. MR 0125307 (23:A2610)
 [3]
 G. Alessandrini, A. Douglis, and E. Fabes, An approximate layering method for the NavierStokes equations in bounded cylinders, Ann. Mat. Pura Appl. 135 (1983), 329347. MR 750540 (86a:35007)
 [4]
 J. T. Beale and A. Majda, Rate of convergence for viscous splitting of the NavierStokes equations, Math. Comp. 37 (1981), 243259. MR 628693 (82i:65056)
 [5]
 G. Benfatto and M. Pulvirenti, Convergence of ChorinMarsden product formula in the halfplane, Comm. Math. Phys. 106 (1986), 427458. MR 859819 (88a:35186)
 [6]
 A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), 785796. MR 0395483 (52:16280)
 [7]
 H. Fujita and T. Kato, On the NavierStokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269315. MR 0166499 (29:3774)
 [8]
 T. Kato, On classical solutions of the twodimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188200. MR 0211057 (35:1939)
 [9]
 O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1969. MR 0254401 (40:7610)
 [10]
 J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, SpringerVerlag, 1972.
 [11]
 F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1968), 329348. MR 0221818 (36:4870)
 [12]
 R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975), 3243. MR 0430568 (55:3573)
 [13]
 L.a. Ying, Viscosity splitting method in bounded domains, SCi. Sinica Ser. A 32 (1989), 908921. MR 1055308 (91h:65201)
 [14]
 L.a. Ying, The viscosity splitting method for the NavierStokes equations in bounded domains, Science Report, Department of Mathematics and Institute of Mathematics, Peking University, October 1986.
 [15]
 , On the viscosity splitting method for initial boundary value problems of the NavierStokes equations, Chinese Ann. Math. 10B (1989), 487512. MR 1038384 (91c:35123)
 [16]
 , Viscosity splitting method for three dimensional NavierStokes equations, Acta Math. Sinica (N.S.) 4 (1988), 210226. MR 965569 (90b:65193)
 [17]
 , Convergence study of viscous splitting in bounded domains, Lecture Notes in Math., vol. 1297, SpringerVerlag, 1987, pp. 184202. MR 929048 (89h:65163)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
35Q30,
65N99,
76D05,
76D07
Retrieve articles in all journals
with MSC:
35Q30,
65N99,
76D05,
76D07
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199010230530
PII:
S 00255718(1990)10230530
Article copyright:
© Copyright 1990
American Mathematical Society
