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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On a one-dimensional $ \alpha$-patch model with nonlocal drift and fractional dissipation


Authors: Hongjie Dong and Dong Li
Journal: Trans. Amer. Math. Soc. 366 (2014), 2041-2061
MSC (2010): Primary 35Q53, 35B44, 35B65
Published electronically: October 23, 2013
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Abstract: We consider a one-dimensional nonlocal nonlinear equation of the form $ \partial _t u = (\Lambda ^{-\alpha } u)\partial _x u - \nu \Lambda ^{\beta }u$, where $ \Lambda =(-\partial _{xx})^{\frac 12}$ is the fractional Laplacian and $ \nu \ge 0$ is the viscosity coefficient. We primarily consider the regime $ 0<\alpha <1$ and $ 0\le \beta \le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $ \alpha $-patch models. In the critical and subcritical range $ 1-\alpha \le \beta \le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $ 0 \le \beta <1-\alpha $, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.


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Additional Information

Hongjie Dong
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email: Hongjie{\textunderscore}Dong@brown.edu

Dong Li
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: mpdongli@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06075-8
PII: S 0002-9947(2013)06075-8
Received by editor(s): July 18, 2012
Published electronically: October 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.