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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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.

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

First available in Project Euclid: 26 April 2012

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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

Navier-Stokes equation bounded mean oscillation dyadic BMO bilinear estimate


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.

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