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Bounded mean oscillation and the uniqueness of active scalar equations


Authors: Jonas Azzam and Jacob Bedrossian
Journal: Trans. Amer. Math. Soc. 367 (2015), 3095-3118
MSC (2010): Primary 35A02; Secondary 35Q92, 35Q35, 76B03
DOI: https://doi.org/10.1090/S0002-9947-2014-06040-6
Published electronically: December 18, 2014
MathSciNet review: 3314802
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Abstract: We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $ L^1 \cap BMO$ and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak-Keller-Segel models with nonlinear diffusion. We obtain these results via technical refinements of energy methods which are well-known in the $ L^2$ setting but are less well-known in the $ \dot {H}^{-1}$ setting. The $ \dot {H}^{-1}$ method can be considered a generalization of Yudovich's classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion and other variants. Important points of our analysis are an $ L^p$-$ BMO$ interpolation lemma and a Sobolev embedding lemma which shows that velocity fields $ v$ with $ \nabla v \in BMO$ are locally log-Lipschitz; the latter is known in harmonic analysis but does not seem to have been connected to this setting.


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Additional Information

Jonas Azzam
Affiliation: Department of Mathematics, University of Washington-Seattle, Seattle, Washington 98195
Address at time of publication: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C Facultat de Ciències, 08193 Bellaterra, Barcelona, Spain
Email: jonasazzam@math.washington.edu, jazzam@mat.uab.cat

Jacob Bedrossian
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Address at time of publication: Department of Mathematics and the Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, Maryland 20742
Email: jacob@cims.nyu.edu, jacob@cscamm.umd.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06040-6
Received by editor(s): August 21, 2011
Received by editor(s) in revised form: November 3, 3012
Published electronically: December 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.