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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  • J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys., 94 (1984), 61–66.
  • R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47 (1984), no.,293.
  • H. Fujita and T. Kato, On the Navier-Stokes initial value problem, I, Arch. Rational Mech. Anal., 16 (1964), 269–315.
  • Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186–212.
  • M. Izuki and Y. Sawano, The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis, J. Math. Anal. Appl., 362 (2010), 140–155.
  • S. Janson, On functions with conditions on the mean oscillation, Ark. Mat., 14 (1976), 189–196.
  • T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $R^{m}$, with applications to weak solutions, Math. Z., 187 (1984), 471–480.
  • G. Koch, N. Nadirashvili, G. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83–105.
  • H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173–194.
  • A. Miyachi, Hardy-Sobolev spaces and maximal functions, J. Math. Soc. Japan, 42 (1990), 73–90.
  • A. Miyachi, Another proof of Kozono-Taniuchi's lemma, 1999, unpublished (in Japanese).
  • T. Sato, Thesis for master's degree (in Japanese), Tohoku University, 2004.
  • Y. Sawano and H. Tanaka, Sharp maximal inequalities and commutators on Morrey spaces with non-doubling measures, Taiwanese J. Math., 11 (2007), 1091–1112.
  • Y. Tsutsui, Sharp maximal inequalities and its application to some bilinear estimates, J. Fourier Anal. Appl., 17 (2011), 265–289.
  • S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa (3), 19 (1965), 593–608.