On a one-dimensional nonlocal flux with fractional dissipation
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 0036-1410Full text not available from this repository.
Official URL: http://dx.doi.org/10.1137/100794924
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|
|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|
|Page Range:||pp. 507-526|
|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)|
 J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the
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