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Viscous splitting for the unbounded problem of the Navier-Stokes equations


Author: Lung-An Ying
Journal: Math. Comp. 55 (1990), 89-113
MSC: Primary 35Q30; Secondary 65N99, 76D05, 76D07
DOI: https://doi.org/10.1090/S0025-5718-1990-1023053-0
MathSciNet review: 1023053
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Abstract: The viscous splitting for the exterior initial-boundary value problems of the Navier-Stokes equations is considered. It is proved that the approximate solutions are uniformly bounded in the space $ {L^\infty }(0,T;{H^{s + 1}}(\Omega ))$, $ s < \frac{3}{2}$, and converge with a rate of $ O(k)$ in the space $ {L^\infty }(0,T;{H^1}(\Omega ))$, where k is the length of the time steps.


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  • [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), 623-727. MR 0125307 (23:A2610)
  • [3] G. Alessandrini, A. Douglis, and E. Fabes, An approximate layering method for the Navier-Stokes equations in bounded cylinders, Ann. Mat. Pura Appl. 135 (1983), 329-347. MR 750540 (86a:35007)
  • [4] J. T. Beale and A. Majda, Rate of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp. 37 (1981), 243-259. MR 628693 (82i:65056)
  • [5] G. Benfatto and M. Pulvirenti, Convergence of Chorin-Marsden product formula in the half-plane, Comm. Math. Phys. 106 (1986), 427-458. MR 859819 (88a:35186)
  • [6] A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), 785-796. MR 0395483 (52:16280)
  • [7] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. MR 0166499 (29:3774)
  • [8] T. Kato, On classical solutions of the two-dimensional non-stationary 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, Gordon and Breach, New York, 1969. MR 0254401 (40:7610)
  • [10] J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Springer-Verlag, 1972.
  • [11] F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1968), 329-348. MR 0221818 (36:4870)
  • [12] R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975), 32-43. MR 0430568 (55:3573)
  • [13] L.-a. Ying, Viscosity splitting method in bounded domains, SCi. Sinica Ser. A 32 (1989), 908-921. MR 1055308 (91h:65201)
  • [14] L.-a. Ying, The viscosity splitting method for the Navier-Stokes 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 Navier-Stokes equations, Chinese Ann. Math. 10B (1989), 487-512. MR 1038384 (91c:35123)
  • [16] -, Viscosity splitting method for three dimensional Navier-Stokes equations, Acta Math. Sinica (N.S.) 4 (1988), 210-226. MR 965569 (90b:65193)
  • [17] -, Convergence study of viscous splitting in bounded domains, Lecture Notes in Math., vol. 1297, Springer-Verlag, 1987, pp. 184-202. MR 929048 (89h:65163)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1023053-0
Article copyright: © Copyright 1990 American Mathematical Society

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