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### On a one-dimensional nonlocal flux with fractional dissipation

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Li, Dong and Rodrigo, Jose L..
(2011)
*On a one-dimensional nonlocal flux with fractional dissipation.*
SIAM Journal on Mathematical Analysis, Volume 43
(Number 1).
pp. 507-526.
ISSN 0036-1410

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

Official URL: http://dx.doi.org/10.1137/100794924

## Abstract

We study a class of one-dimensional conservation laws with nonlocal flux and fractional dissipation: partial derivative(t)theta - (theta H theta)(x) = -nu(-partial derivative(xx))(gamma/2)theta, where H is the Hilbert transform. In the regime nu > 0 and 1 < gamma <= 2, we prove local existence and regularity of solutions regardless of the sign of the initial data. For all values nu >= 0 and 0 <= gamma <= 2, we construct a certain class of positive smooth initial data with sufficiently localized mass, such that corresponding solutions blow up in finite time. This extends recent results of Castro and Cordoba [Adv. Math., 219 (2008), pp. 1916-1936].

Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Hilbert transform, Fluid mechanics, One-dimensional flow |

Journal or Publication Title: | SIAM Journal on Mathematical Analysis |

Publisher: | Society for Industrial and Applied Mathematics |

ISSN: | 0036-1410 |

Official Date: | 2011 |

Volume: | Volume 43 |

Number: | Number 1 |

Page Range: | pp. 507-526 |

Identification Number: | 10.1137/100794924 |

Status: | Peer Reviewed |

Publication Status: | Published |

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

Funder: | National Science Foundation (U.S.) (NSF), University of Iowa. Department of Mathematics, University of Iowa, Spain. Ministerio de Educación y Ciencia (MEC) |

Grant number: | 0908032 (NSF), MTM2005-05980-C02-01 (MEC) |

References: | [1] J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the |

URI: | http://wrap.warwick.ac.uk/id/eprint/41510 |

Data sourced from Thomson Reuters' Web of Knowledge

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