Advertisement

Existence proof of unimodal solutions of the Proudman–Johnson equation via interval analysis

  • Tomoyuki MiyajiEmail author
  • Hisashi Okamoto
Original Paper Area 1
  • 23 Downloads

Abstract

The Proudman–Johnson equation is considered as a representative of the two-dimensional fluid flow. Its steady-states are computed and their existence and unimodality are proved rigorously via the interval analysis and the interval Newton method. By using both the multiple shooting method and multiple-precision arithmetic our verification succeeds up to the Reynolds number \(\le 5000\).

Keywords

The Proudman–Johnson equation Unimodal solutions Interval analysis Computer-assisted proof Multiple shooting method 

Mathematics Subject Classification

35Q35 76D99 65G20 

Notes

Acknowledgements

The present work was partially supported by JSPS A3 Foresight Program. It is a pleasure of the second author to acknowledge the support from the Erwin Schrödinger Institute, Vienna, Austria.

References

  1. 1.
    Bae, H., Chae, D., Okamoto, H.: On the well-posedness of various one-dimensional model equations for fluid motion. Nonlinear Anal. 160, 25–43 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  3. 3.
    Jones, E., Oliphant, T., Peterson, P., and others, SciPy: Open source scientific tools for Python, 2001. http://www.scipy.org/. Last accessed 3 Aug 2018
  4. 4.
    Kim, S.-C., Miyaji, T., Okamoto, H.: Unimodal patterns appearing in the two-dimensional Navier–Stokes flows under general forcing at large Reynolds numbers. Eur. J. Mech. B Fluids 65, 234–246 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kim, S.-C., Okamoto, H.: Vortices of large scale appearing in the 2D stationary Navier–Stokes equations at large Reynolds numbers. Jpn. J. Ind. Appl. Math. 27, 47–71 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kim, S.-C., Okamoto, H.: The generalized Proudman–Johnson equation at large Reynolds numbers. IMA J. Appl. Math. 78, 379–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kim, S.-C., Okamoto, H.: Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers. Nonlinearity 28, 3219–3242 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kraichnan, R.: Inertial ranges in two-dimensional turbulence. Phys. Fluid 10, 1417–1423 (1967)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lohner, R.J.: Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems. In: Cash, J.R., Gladwell, I. (eds.) Computational Ordinary Differential Equations. Clarendon Press, Oxford (1992)Google Scholar
  10. 10.
    Miyaji, T.: Tomoyuki Miyaji’s web page. http://www.kisc.meiji.ac.jp/~tmiyaji/. Last accessed 12 Sep 2018
  11. 11.
    Miyaji, T., Okamoto, H.: A computer-assisted proof of existence of a periodic solution. Proc. Jpn. Acad. Ser. A 90, 139–144 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miyaji, T., Okamoto, H., Craik, A.D.D.: A four-leaf chaotic attractor of a three-dimensional dynamical system. Int. J. Bifur. Chaos. 25, 1530003 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Morrison, D.D., Riley, J.D., Zancanaro, J.F.: Multiple shooting method for two-point boundary value problems. Commun. ACM 5, 613–614 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Okamoto, H.: Models and Special Solutions of the Navier–Stokes Equations. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, New York (2018)Google Scholar
  15. 15.
    Okamoto, H., Zhu, J.: Some similarity solutions of the Navier–Stokes equations and related topics. Taiwan. J. Math. 4, 65–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Proudman, I., Johnson, K.: Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rump, S.M.: Verification methods: Rigorous results using floating-point arithmetic. Acta Num. 19, 287–449 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schwartz, I.B.: Estimating regions of existence of unstable periodic orbits using computer-based techniques. SIAM J. Num. Anal. 20, 106–120 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Valenca, M.R.: Multiple shooting using interval analysis. BIT 25, 425–427 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    van Dyke, M.: An Album of Fluid Motion. Parabolic Press, Stanford (1982)Google Scholar
  21. 21.
    Watanabe, Y.: A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math. 223, 953–966 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Watanabe, Y.: An efficient numerical verification method for the Kolmogorov problem of incompressible viscous fluid. J. Comput. Appl. Math. 302, 157–170 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wilczak, D. et al.: Computer Assisted Proofs in Dynamics, Jagiellonian University. http://capd.ii.uj.edu.pl/. Last accessed 3 Aug 2018
  24. 24.
    Zgliczynski, P.: \(C^1\) Lohner algorithm. Found. Comput. Math. 2, 429–465 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan
  2. 2.Department of MathematicsGakushuin UniversityTokyoJapan

Personalised recommendations