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

  • Tomoyuki MiyajiEmail author
  • Hisashi Okamoto
Original Paper Area 1


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 \(\le 5000\).


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

Mathematics Subject Classification

35Q35 76D99 65G20 



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.


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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

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