The Initial Boundary Value Problem for Navier-Stokes Equations
Acta Mathematica Sinica, English Series
Acta Mathematica Sinica,
Chinese Series
Adv Search »  
Acta Mathematica Sinica, English Series  1999, Vol. 15 Issue (2): 153-164    DOI: 10.1007/BF02650658
articles Current Issue | Next Issue | Archive | Adv Search  |   
The Initial Boundary Value Problem for Navier-Stokes Equations
Cheng He
Institute of Applied Mathmatics, Academia Sinica, Beijing 100080, P. R. China
 Download: PDF (180 KB)   HTML (0 KB)   Export: BibTeX | EndNote (RIS)      Supporting Info
Abstract By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformly C3 boundary, under the assumption that‖aL 2(Θ)+‖fL 1(0,∞;L2(Θ)) or‖▽aL 2(Θ)+‖f|L 2(0,∞;L2(Θ)) small or viscosityv large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.
E-mail this article
Add to my bookshelf
Add to citation manager
E-mail Alert
Articles by authors
Cheng He
Key wordsNavier-Stokes equations   Stokes equations   Homogeneous boundary conditions   Nonhomogeneous boundary conditions     
Received: 1994-04-07;
Fund: This work is supported by foundation of Institute of Mathematics, Academia Sinica
Cite this article:   
Cheng He. The Initial Boundary Value Problem for Navier-Stokes Equations[J]. Acta Mathematica Sinica, English Series, 1999, 15(2): 153-164.
URL:      or
[1] A J Chorin,J E Marsdon.A Mathematical Introduction to Fluid Mechanics.Second Edition,Springer-Verlag,1990
[2] J Leray.Sur le mouvement d'un liquide visqueux emplissant l'espace.Acta Math,1934,63(2):193-248
[3] J L Lions,G Prodi.Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2.C R Acad Sci Paris,1959,248:3519-3521.
[4] O A Ladyzhenskaya.Solution “in the large” of the nonstationary boundary value problem for the Navier-Stokes system with two space variables.Comm Pure Appl Math,1959,12(4):427-433
[5] E Hopf.über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen.Math Nachr,1951,4(4):213-231
[6] H Fujita,T Kato.On the Navier-Stokes initial value problem I.Arch Rat Mech Anal,1964,16(4):269-315
[7] Y Giga.Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system.J Diff Equation,1986,61(2):186-212
[8] Y Giga,T Miyakawa.Solutions in Lr to the Navier-Stokes initial value problem.Arch Rat Mech Anal,1985,89(3):267-281
[9] J G Heywood.The Navier-Stokes equations:On the existence,regularity and decay of solutions.Indiana Univ Math J,1980,29(5):639-681
[10] S Ito.The existence and the uniqueness of regular solution of nonstationary Navier-Stokes equations.J Fac Sci Univ Tokyo Sect IA,1961,9(2):103-140
[11] A A Kiselev,O A Ladyzhenskaya.On the existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid.Izv Akad Nauk SSSR Ser Mat,1957,21(4):655-680
[12] O A Ladyzhenskaya.The Mathematical Theory of Viscous Incompressible Flow.Gordon and Breach,1969
[13] A Mahalov,E S Titi,S Leibovich.Invariant helical subspaces for the Navier-Stokes equations.Arch Rat Mech Anal,1990,112(2):193-222
[14] T Miyakawa.On nonstationary solutions of the Navier-Stokes equations in an exterior domain.Hiroshima Math J,1982,12(1):115-140
[15] G Prodi.Un teorema di unicità per le equazioni di Navier-Stokes.Ann Mat Pura Appl,1959,48(1):173-182
[16] J Serrin.The initial value problem for the Navier-Stokes equations.Nonlinear Problems.Edited by R E Langer.The Univ of Wisconsin Press,Madison,1963,69-98
[17] M Sinbrot,S Kaniel.The initial value problem for the Navier-Stokes equations.Arch Rat Mech Anal,1966,21(4):270-285
[18] M R Ukhovskii,V I Iudovich.Axially symmetric flows of ideal and viscous fluids filling the whole space.J Appl Math Mech,1968,32(1):52-62
[19] B F Weissler.The Navier-Stokes initial value problem in Lp.Arch Rat Mech Anal,1980,74(3):219-230
[20] T Kato.Strong Lp-solutions of the Navier-Stokes Equations in Rn,with Application to weak solutions.Math Z,1984,187(4):471-480
[21] S Ukai.A solution formula for the Stokes equation in R+n .Comm Pure Appl Math,1987,40(5):611-621
[22] H Iwashita.Lp-Lq estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problem in Lq space.Math Ann,1989,285(2):265-288
[23] H Kozono,T Ogawa.Global strong solution and its decay properties for the Navier-stokes equations in three dimensional domains with non-compact boundaries.Math Z,1994,216(1):1-30
[24] G Ponce,R Racke,T C Sideris,E S Titi.Global stability of large solutions to the 3D-Navier-Stokes equations.Comm Math Phys,1994,159(2):320-341
[25] R Temam.Navier-Stokes Equations.North-Holland Publ co,1977
No Similar of article
  Copyright 2012 © Editorial Office of Acta Mathematica Sinica