Journal of the Mathematical Society of Japan

Bilinear estimates in dyadic BMO and the Navier-Stokes equations

Eiichi NAKAI and Tsuyoshi YONEDA

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Abstract

We establish bilinear estimates in dyadic BMO as an extension of Kozono and Taniuchi's result on the usual BMO. To establish the bilinear estimates we use sharp maximal functions, while they used the boundedness of pseudo-differential operators by Coifman and Meyer. By this extension we prove that the dyadic BMO norm of the velocity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Moreover, we give an odd function with type II singularity which clarifies the difference between BMO and dyadic BMO.

Article information

Source
J. Math. Soc. Japan Volume 64, Number 2 (2012), 399-422.

Dates
First available in Project Euclid: 26 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1335444398

Digital Object Identifier
doi:10.2969/jmsj/06420399

Mathematical Reviews number (MathSciNet)
MR2916074

Zentralblatt MATH identifier
1251.35073

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 42B35: Function spaces arising in harmonic analysis

Keywords
Navier-Stokes equation bounded mean oscillation dyadic BMO bilinear estimate

Citation

NAKAI, Eiichi; YONEDA, Tsuyoshi. Bilinear estimates in dyadic BMO and the Navier-Stokes equations. J. Math. Soc. Japan 64 (2012), no. 2, 399--422. doi:10.2969/jmsj/06420399. https://projecteuclid.org/euclid.jmsj/1335444398.


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