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On a onedimensional nonlocal flux with fractional dissipation
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Li, Dong and Rodrigo, Jose L.. (2011) On a onedimensional nonlocal flux with fractional dissipation. SIAM Journal on Mathematical Analysis, Volume 43 (Number 1). pp. 507526. ISSN 00361410
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Official URL: http://dx.doi.org/10.1137/100794924
Abstract
We study a class of onedimensional 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. 19161936].
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Hilbert transform, Fluid mechanics, Onedimensional flow  
Journal or Publication Title:  SIAM Journal on Mathematical Analysis  
Publisher:  Society for Industrial and Applied Mathematics  
ISSN:  00361410  
Official Date:  2011  
Dates: 


Volume:  Volume 43  
Number:  Number 1  
Page Range:  pp. 507526  
Identifier:  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), MTM200505980C0201 (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 
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