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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 3(375), Pages 47–72 (Mi umn6760)  

This article is cited in 6 scientific papers (total in 6 papers)

Orthonormal quaternion frames, Lagrangian evolution equations, and the three-dimensional Euler equations

J. Gibbon

Imperial College, Department of Mathematics

Abstract: More than 160 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the orientation and paths of moving objects undergoing three-axis rotations. Here it is shown that they provide a natural way of selecting an appropriate orthonormal frame — designated the quaternion-frame — for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has some bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale–Kato–Majda theorem and associated work on the direction of vorticity by Constantin, Fefferman, and Majda and by Deng, Hou, and Yu. It is then shown how the quaternion formalism provides an alternative formulation in terms of the Hessian of the pressure.

DOI: https://doi.org/10.4213/rm6760

Full text: PDF file (846 kB)
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English version:
Russian Mathematical Surveys, 2007, 62:3, 535–560

Bibliographic databases:

UDC: 517.958+531.3-322
MSC: Primary 35Q35; Secondary 35B40, 35L60, 35Q30, 46N20, 76B03, 76B47, 76D05, 7
Received: 27.09.2006

Citation: J. Gibbon, “Orthonormal quaternion frames, Lagrangian evolution equations, and the three-dimensional Euler equations”, Uspekhi Mat. Nauk, 62:3(375) (2007), 47–72; Russian Math. Surveys, 62:3 (2007), 535–560

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Gibbon J.D., “The three-dimensional Euler equations: Where do we stand?”, Phys. D, 237:14-17 (2008), 1894–1904  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Eshraghi H., Gibbon J.D., “Quaternions and ideal flows”, J. Phys. A, 41:34 (2008), 344004, 19 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    3. Gibbon J.D., Bustamante M., Kerr R.M., “The three-dimensional Euler equations: singular or non-singular?”, Nonlinearity, 21:8 (2008), T123–T129  crossref  mathscinet  zmath  isi  scopus
    4. Roulstone I., Banos B., Gibbon J.D., Roubtsov V.N., “A geometric interpretation of coherent structures in Navier–Stokes flows”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465:2107 (2009), 2015–2021  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Bustamante M.D., “3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis”, Phys. D, 240:13 (2011), 1092–1099  crossref  mathscinet  zmath  isi  scopus
    6. Ohkitani K., “Dynamical Equations For the Vector Potential and the Velocity Potential in Incompressible Irrotational Euler Flows: a Refined Bernoulli Theorem”, 92, no. 3, 2015, 033010  crossref  isi  scopus
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