# The Library

### The three-dimensional Euler equations : singular or non-singular?

Tools

Gibbon, J. D., Bustamante, Miguel D. and Kerr, Robert M. (Robert McDougall), 1954-.
(2008)
*The three-dimensional Euler equations : singular or non-singular?*
Nonlinearity, Volume 21
(Number 8).
T123-T129.
ISSN 0951-7715

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1088/0951-7715/21/8/T02

## Abstract

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Engineering Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Time-series analysis, Symmetry (Physics), Singularities (Mathematics), Mathematical models, Turbulence, Fluid dynamics, Equations of motion |

Journal or Publication Title: | Nonlinearity |

Publisher: | Institute of Physics Publishing Ltd. |

ISSN: | 0951-7715 |

Date: | August 2008 |

Volume: | Volume 21 |

Number: | Number 8 |

Number of Pages: | 7 |

Page Range: | T123-T129 |

Identification Number: | 10.1088/0951-7715/21/8/T02 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

References: | [1] Majda A J and Bertozzi A L 2001 Vorticity and Incompressible Flow (Cambridge: Cambridge University Press) [2] Constantin P 2008 Proc. Conf. “Euler Equations 250 Years On” (Aussois, France, June 2007) Physica D to appear August 2008 [3] Morf R H, Orszag S A and Frisch U 1980 Spontaneous singularity in three-dimensional, inviscid incompressible flow Phys. Rev. Lett. 44 572–5 [4] Bardos C, Benachour S and Zerner M 1976 Analyticit´e des solutions p´eriodiques de l’´equation d’Euler en deux dimensions C. R. Acad. Sci. Paris 282A 995–8 [5] Bardos C and Benachour S 1977 Domaine d’analyticit´e des solutions de l’´equation d’Euler dans un ouvert de Rn Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV Ser. 4 647–87 [6] Pauls W, Matsumoto T, Frisch U and Bec J 2006 Nature of complex singularities for the 2D Euler equation Physica D 219 40–59 [7] Chorin A J 1982 The evolution of a turbulent vortex Commun. Math. Phys. 83 517–35 [8] Brachet M E, Meiron D I, Orszag S A, Nickel B G, Morf R H and Frisch U 1983 Small-scale structure of the Taylor–Green vortex J. Fluid Mech. 130 411–52 [9] Siggia E D 1984 Collapse and amplification of a vortex filament Phys. Fluids 28 794–805 [10] Ashurst W and Meiron D 1987 Numerical study of vortex reconnection Phys. Rev. Lett. 58 1632–5 [11] Pumir A and Kerr R M 1987 Numerical simulation of interacting vortex tubes Phys. Rev. Lett. 58 1636–39 [12] Pumir A and Siggia E 1990 Collapsing solutions to the 3D Euler equations, Phys. Fluids A 2 220–41 [13] Bell J B and Marcus D L 1992 Vorticity intensification and transition to turbulence in the three-dimensional Euler equations Commun. Math. Phys. 147 371–94 [14] Brachet M E, Meneguzzi V, Vincent A, Politano H and Sulem P-L 1992 Numerical evidence of smooth selfsimilar dynamics and the possibility of subsequent collapse for ideal flows Phys. Fluids A 4 2845–54 [15] Kerr RM1993 Evidence for a singularity of the three-dimensional incompressible Euler equations Phys. Fluids A 5 1725–46 [16] Kerr R M 2005 Vorticity and scaling of collapsing Euler vortices Phys. Fluids A 17 075103–114 [17] BustamanteMD and Kerr RM2008 3D Euler about a 2D symmetry plane Proc. of “Euler Equations 250 Years On” (Aussois, France, June 2007) Physica D to appear August 2008 (doi:10.1016/j.physd.2008.02.007) [18] Hou T Y and Li R 2006 Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler Equations J. Nonlinear Sci. 16 639–64 [19] Hou T Y and Li R 2008 Blowup or no blowup? The interplay between theory and numerics Proc. “Euler Equations 250 Years On” (Aussois, France, June 2007) Physica D to appear August 2008 [20] Orlandi P and Carnevale G 2007 Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles Phys. Fluids 19 057106 [21] Grauer R, Marliani C and Germaschewski K 1998 Adaptive mesh refinement for singular solutions of the incompressible Euler equations Phys. Rev. Lett. 80 4177–80 [22] Grafke T, Homann H, Dreher J and Grauer R 2008 Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods Proc. “Euler Equations 250 Years On” (Aussois, France, June 2007) Physica D to appear August 2008 [23] Kida S 1985 Three-dimensional periodic flows with high-symmetry J. Phys. Soc. Japan 54 2132–6 [24] Kerr R M 2005 Vortex collapse and turbulence Fluid Dyn. Res. 36 249–60 [25] Boratav O N and Pelz R B 1994 Direct numerical simulation of transition to turbulence from a high-symmetry initial condition Phys. Fluids 6 2757–84 [26] Boratav O N and Pelz R B 1995 On the local topology evolution of a high-symmetry flow Phys. Fluids 7 1712–31 [27] Pelz R B 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model Phys. Rev. E 55 1617–26 [28] Pelz R B and Gulak Y 1997 Evidence for a real-time singularity in hydrodynamics from time series analysis Phys. Rev. Lett. 79 4998–5001 [29] Pelz R B 2001 Symmetry and the hydrodynamic blow-up problem J. Fluid Mech. 444 299–320 [30] Cichowlas C and Brachet M-E 2005 Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows Fluid Dyn. Res. 36 239–48 [31] Gulak Y and Pelz R B 2005 High-symmetry Kida flow: time series analysis and resummation Fluid Dyn. Res. 36 211–20 [32] Pelz R B and Ohkitani K 2005 Linearly strained flows with and without boundaries—the regularizing effect of the pressure term Fluid Dyn. Res. 36 193–210 [33] Gibbon J D 2008 The three dimensional Euler equations: how much do we know? Proc. “Euler Equations 250 Years On” (Aussois, France, June 2007) PhysicaDto appear August 2008 (doi:10.1016/j.physd.2007.10.014) [34] Bardos C and Titi E S 2007 Euler equations of incompressible ideal fluids Russ. Math. Surv. 62 409–51 [35] Beale J T, Kato T and Majda A J 1984 Remarks on the breakdown of smooth solutions for the 3D Euler equations Commun. Math. Phys. 94 61–6 [36] Ferrari A 1993 On the blow-up of solutions of the 3D Euler equations in a bounded domain Commun. Math. Phys. 155 277–94 [37] Kozono H and Taniuchi Y 2000 Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations Commun. Math. Phys. 214 191–200 [38] Ponce G 1985 Remarks on a paper by J. T. Beale, T. Kato and A. Majda Commun. Math. Phys. 98 349–53 [39] Chae D 2003 Remarks on the blow-up of the Euler equations and the related equations Commun. Math. Phys. 245 539–50 [40] Chae D 2004 Local existence and blow-up criterion for the Euler equations in the Besov spaces Asymptotic Analysis 38 339–58 [41] Chae D 2005 Remarks on the blow-up criterion of the 3D Euler equations Nonlinearity 18 1021–9 [42] Chae D 2007 On the finite time singularities of the 3D incompressible Euler equations Commun. Pure Appl. Math. 60 597–617 [43] Constantin P 1994 Geometric statistics in turbulence SIAM Rev. 36 73–98 [44] Constantin P, Fefferman Ch and Majda A J 1996 Geometric constraints on potentially singular solutions for the 3D Euler equation Commun. Partial Diff. Eqns 21 559–71 [45] Cordoba D and Fefferman Ch 2001 On the collapse of tubes carried by 3D incompressible flows Commun. Math. Phys. 222 293–8 [46] Deng J, Hou T Y and Yu X 2005 Geometric properties and non-blowup of 3D incompressible Euler flow Commun. Partial Diff. Eqns 30 225–43 [47] Deng J, Hou T Y and Yu X 2003 Improved geometric condition for non-blowup of the 3D incompressible Euler equation Commun. Partial Diff. Eqns 31 293–306 [48] Gibbon J D 2007 Ortho-normal quaternion frames, Lagrangian evolution equations and the three-dimensional Euler equations Russ. Math. Surv. 62 1–26 Gibbon J D 2007 Uspekhi Mat. Nauk 62 47–72 [49] Constantin P and Foias C 1988 Navier–Stokes Equations (Chicago, IL: The University of Chicago Press) [50] Foias C, Manley O, Rosa R and Temam R 2001 Navier–Stokes equations & Turbulence (Cambridge: Cambridge University Press) [51] Brenier Y 1999 Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations Commun. Pure Appl. Math. 52 411–52 [52] Shnirelman A 1997 On the non-uniqueness of weak solution of the Euler equation Commun. Pure Appl. Math. 50 1260–86 [53] De Lellis C and Sz´ekelyhidi L 2007 The Euler equations as a differential inclusion Ann. Math. to be published [54] Gibbon J D and Doering C R 2003 Intermittency is solutions of the three-dimensional Navier–Stokes equations J. Fluid Mech. 478 227–35 [55] Gibbon J D and Doering 2005 Intermittency & regularity issues in three-dimensional Navier–Stokes turbulence Arch. Rat. Mech. Anal. 177 115–50 [56] Gibbon J D and Titi E S 2005 Cluster formation in complex multi-scale systems Proc. R. Soc. 461 3089–97 [57] Gibbon J D and Pavliotis G A 2007 Estimates for the two-dimensional Navier–Stokes equations in terms of the Reynolds number J. Math. Phys. 48 065202 |

URI: | http://wrap.warwick.ac.uk/id/eprint/29706 |

Data sourced from Thomson Reuters' Web of Knowledge

### Actions (login required)

View Item |