Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T15:58:59.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Meromorphic non-integrability of a steady Stokes flow inside a sphere

Published online by Cambridge University Press:  15 November 2012

TAKAHIRO NISHIYAMA
TAKAHIRO NISHIYAMA*
Affiliation:
Department of Applied Science, Yamaguchi University, Ube 755-8611, Japan (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The non-existence of a real meromorphic first integral for a spherically confined steady Stokes flow of Bajer and Moffatt is proved on the basis of Ziglin’s theory and the differential Galois theory. In the proof, the differential Galois group of a second-order Fuchsian-type differential equation associated with normal variations along a particular streamline is shown to be a special linear group according to Kovacic’s algorithm. A set of special values of a parameter contained in the Fuchsian-type equation is studied by using the theory of elliptic curves. For this set, a computer algebra system is used in part of Kovacic’s algorithm.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 616 - 627
Copyright
©2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bajer, K. and Moffatt, H. K.. On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212 (1990), 337–363.Google Scholar
[2]Chahal, J. S.. Topics in Number Theory. Plenum, New York, 1988.Google Scholar
[3]Churchill, R. C. and Rod, D. L.. On the determination of Ziglin monodromy groups. SIAM J. Math. Anal. 22 (1991), 1790–1802.Google Scholar
[4]Duval, A. and Loday-Richaud, M.. Kovacic’s algorithm and its application to some families of special functions. Appl. Algebra Engrg. Comm. Comput. 3 (1992), 211–246.Google Scholar
[5]Kovacic, J. J.. An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1986), 3–43.CrossRefGoogle Scholar
[6]Maciejewski, A. J. and Przybylska, M.. Non-integrability of ABC flow. Phys. Lett. A 303 (2002), 265–272.Google Scholar
[7]Morales Ruiz, J. J.. Differential Galois Theory and Non-Integrability of Hamiltonian Systems. Birkhäuser, Basel, 1999.Google Scholar
[8]Morales, J. J. and Simó, C.. Picard–Vessiot theory and Ziglin’s theorem. J. Differential Equations 107 (1994), 140–162.Google Scholar
[9]Ono, T.. Variations on a Theme of Euler: Quadratic Forms, Elliptic Curves, and Hopf Maps. Plenum, New York, 1994.Google Scholar
[10]Vainshtein, D. L., Vasiliev, A. A. and Neishtadt, A. I.. Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow. Chaos 6 (1996), 67–77.Google Scholar
[11]Ziglin, S. L.. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I. Funct. Anal. Appl. 16 (1982), 181–189.CrossRefGoogle Scholar
[12]Ziglin, S. L.. On the absence of a real-analytic first integral for ABC flow when A=B. Chaos 8 (1998), 272–273.Google Scholar