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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations


Authors: Chris Orum and Mina Ossiander
Journal: Proc. Amer. Math. Soc. 141 (2013), 3883-3897
MSC (2010): Primary 35Q30, 76D05, 42B37; Secondary 76M35, 39B72, 60J80
Published electronically: July 30, 2013
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Abstract: The convolution inequality $ h*h(\xi ) \leq B \vert\xi \vert^\theta h(\xi )$ defined on $ \mathbf {R}^n$ arises from a probabilistic representation of solutions of the $ n$-dimensional Navier-Stokes equations, $ n \geq 2$. Using a chaining argument, we establish in all dimensions $ n \geq 1$ the nonexistence of strictly positive fully supported solutions of this inequality for $ \theta \geq n/2.$ We use this result to describe a chain of continuous embeddings from spaces associated with probabilistic solutions to the spaces $ BMO^{-1}$ and $ BMO_T^{-1}$ associated with the Koch-Tataru solutions of the Navier-Stokes equations.


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Additional Information

Chris Orum
Affiliation: Department of Mathematics, University of Utah, 155 S. 1400 E., Room 233, Salt Lake City, Utah 84112-0090
Address at time of publication: Mathematics Program, Eastern Oregon University, La Grande, Oregon 97850
Email: orum@math.utah.edu, orum@math.utah.edu

Mina Ossiander
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: ossiand@math.oregonstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11662-X
PII: S 0002-9939(2013)11662-X
Received by editor(s): January 13, 2011
Received by editor(s) in revised form: January 22, 2012
Published electronically: July 30, 2013
Additional Notes: This work was partially supported by the U.S. National Science Foundation through the Focussed Research Group collaborative awards DMS-0073958 and DMS-0940249
Communicated by: Walter Craig
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.