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### An example of finite-time singularities in the 3d Euler equations

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He, Xinyu.
(2007)
*An example of finite-time singularities in the 3d Euler equations.*
Journal of Mathematical Fluid Mechanics, Vol.9
(No.3).
pp. 398-410.
ISSN 1422-6928

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

Official URL: http://dx.doi.org/10.1007/s00021-005-0205-3

## Abstract

Let Omega = R-3\(B) over bar _1(0) be the exterior of the closed unit ball. Consider the self-similar Euler system

alpha u+beta y.del u+u.del u+del p=0, div u=0 in Omega.

Setting alpha=beta=1/2 gives the limiting case of Leray's self-similar Navier-Stokes equations. Assuming smoothness and smallness of the boundary data on partial derivative Omega, we prove that this system has a unique solution (u,p) epsilon C-1(Omega;R(3)xR), vanishing at infinity, precisely

u(y)down arrow 0 as vertical bar y vertical bar up arrow infinity, with u = O(vertical bar y vertical bar(-1)), del u=O(vertical bar y vertical bar(-2)).

The self-similarity transformation is v(x,t)=u(y)/(t*-t)(alpha), y=x/(t*-t)(beta), where v(x,t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x,t) blows up at (x*,t*), x*=0, t*<+infinity. This isolated singularity has bounded energy with unbounded L-2-norm of curl v.

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

Subjects: | Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery Q Science > QC Physics |

Divisions: | Faculty of Science > Mathematics |

Journal or Publication Title: | Journal of Mathematical Fluid Mechanics |

Publisher: | Birkhaeuser Verlag AG |

ISSN: | 1422-6928 |

Official Date: | August 2007 |

Volume: | Vol.9 |

Number: | No.3 |

Number of Pages: | 13 |

Page Range: | pp. 398-410 |

Identification Number: | 10.1007/s00021-005-0205-3 |

Status: | Peer Reviewed |

Publication Status: | Published |

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

URI: | http://wrap.warwick.ac.uk/id/eprint/31610 |

Data sourced from Thomson Reuters' Web of Knowledge

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