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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. doi:10.1137/100794924 ISSN 0036-1410.
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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, Engineering and Medicine > 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 | ||||
Dates: |
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Volume: | Volume 43 | ||||
Number: | Number 1 | ||||
Page Range: | pp. 507-526 | ||||
DOI: | 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) |
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