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- Publisher Website: 10.1090/S0002-9947-2013-06075-8
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Article: On a one-dimensional α-patch model with nonlocal drift and fractional dissipation
Title | On a one-dimensional α-patch model with nonlocal drift and fractional dissipation |
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Authors | |
Issue Date | 2014 |
Citation | Transactions of the American Mathematical Society, 2014, v. 366, n. 4, p. 2041-2061 How to Cite? |
Abstract | We consider a one-dimensional nonlocal nonlinear equation of the form ∂tu = (Λ- αu)dxu-vΛ βu, where Λ = (-∂xx)1/2 is the fractional Laplacian and v > 0 is the viscosity coefficient. We primarily consider the regime 0 < α, < 1 and 0 < β < 2 for which the model has no local drift, fractional dissipation, and captures essential features of the 2D a-patch models. In the critical and sub critical range 1 - α, < β < 2, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 < β < 1 - α, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonsocial weighted inequality which can be of independent interest. © 2013 American Mathematical Society Reverts to public domain 28 years from publication. |
Persistent Identifier | http://hdl.handle.net/10722/326976 |
ISSN | 2023 Impact Factor: 1.2 2023 SCImago Journal Rankings: 1.581 |
DC Field | Value | Language |
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dc.contributor.author | Dong, Hongjie | - |
dc.contributor.author | Li, Dong | - |
dc.date.accessioned | 2023-03-31T05:27:53Z | - |
dc.date.available | 2023-03-31T05:27:53Z | - |
dc.date.issued | 2014 | - |
dc.identifier.citation | Transactions of the American Mathematical Society, 2014, v. 366, n. 4, p. 2041-2061 | - |
dc.identifier.issn | 0002-9947 | - |
dc.identifier.uri | http://hdl.handle.net/10722/326976 | - |
dc.description.abstract | We consider a one-dimensional nonlocal nonlinear equation of the form ∂tu = (Λ- αu)dxu-vΛ βu, where Λ = (-∂xx)1/2 is the fractional Laplacian and v > 0 is the viscosity coefficient. We primarily consider the regime 0 < α, < 1 and 0 < β < 2 for which the model has no local drift, fractional dissipation, and captures essential features of the 2D a-patch models. In the critical and sub critical range 1 - α, < β < 2, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 < β < 1 - α, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonsocial weighted inequality which can be of independent interest. © 2013 American Mathematical Society Reverts to public domain 28 years from publication. | - |
dc.language | eng | - |
dc.relation.ispartof | Transactions of the American Mathematical Society | - |
dc.title | On a one-dimensional α-patch model with nonlocal drift and fractional dissipation | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1090/S0002-9947-2013-06075-8 | - |
dc.identifier.scopus | eid_2-s2.0-84892988436 | - |
dc.identifier.volume | 366 | - |
dc.identifier.issue | 4 | - |
dc.identifier.spage | 2041 | - |
dc.identifier.epage | 2061 | - |