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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1213–1228 | Cite as

Stability of Bifurcating Stationary Solutions of the Artificial Compressible System

  • Yuka Teramoto
Article

Abstract

The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number \(\epsilon \) which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small \(\epsilon \). In general, the range of \(\epsilon \) shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of \(\epsilon \) can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

Keywords

Incompressible Navier–Stokes system Artificial compressible system Singular perturbation Stability Bifurcation 

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Notes

Acknowledgements

I am deeply grateful to Professor Yoshiyuki Kagei for his constant encouragement. This work was partly supported by JSPS KAKENHI Grant Number 17J04702.

Compliance with ethical standards

Conflict of interest

The author declares that she has no conflict of interest.

References

  1. 1.
    Amann, H.: Ordinary differential equations. An introduction to nonlinear analysis. Translated from the German by Gerhard Metzen, De Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter & Co., Berlin (1990)Google Scholar
  2. 2.
    Chorin, A.: The numerical solution of the Navier–Stokes equations for an incompressible fluid. Bull. Am. Math. Soc. 73, 928–931 (1967)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chorin, A.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Chorin, A.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Crandall, M., Rabinowitz, P.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Donatelli, D.: On the artificial compressibility method for the Navier–Stokes–Fourier system. Q. Appl. Math. 68, 469–485 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Donatelli, D.: The artificial compressibility approximation for MHD equations in unbounded domain. J. Hyperb. Differ. Equ. 10, 181–198 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Donatelli, D., Marcati, P.: A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D. J. Hyperb. Differ. Equ. 3, 575–588 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Donatelli, D., Marcati, P.: Leray weak solutions of the incompressible Navier–Stokes system on exterior domains via the artificial compressibility method. Indiana Univ. Math. J. 59, 1831–1852 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. 1. Springer, New York (1994)MATHGoogle Scholar
  11. 11.
    Kagei, Y., Nishida, T.: On Chorin’s method for stationary solutions of the Oberbeck–Boussinesq equation. J. Math. Fluid Mech. 19, 345–365 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kagei, Y., Nishida, T., Teramoto, Y.: On the spectrum for the artificial compressible system. J. Differ. Equ. 264, 897–928 (2018)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kirchgässner, K., Sorger, P.: Branching analysis for the Taylor problem. Q. J. Mech. Appl. Math. 22, 183–209 (1969)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. I. Arch. Ration. Mech. Anal. 32, 135–153 (1969)CrossRefMATHGoogle Scholar
  15. 15.
    Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)CrossRefMATHGoogle Scholar
  16. 16.
    Temam, R.: Navier–Stokes equations. Theory and numerical analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI (2001)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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