Hokkaido Mathematical Journal

Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations

Eiichi NAKAI and Tsuyoshi YONEDA

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Abstract

The purpose of this paper is twofold. Let Rj (j = 1,2, ... , n) be Riesz transforms on $\mathbb{R}$n. First we prove the convergence of truncated operators of RiRj in generalized Hardy spaces. Our first result is an extension of the convergence in Lp($\mathbb{R}$^n) (1 < p < ∞). Secondly, as an application of the first result, we show a uniqueness theorem for the Navier-Stokes equation. J. Kato (2003) established the uniqueness of solutions of the Navier-Stokes equations in the whole space when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. We extend the part "BMO-valued" in his result to "generalized Campanato space valued". The generalized Campanato spaces include L1, BMO and homogeneous Lipschitz spaces of order α (0 < α < 1).

Article information

Source
Hokkaido Math. J., Volume 40, Number 1 (2011), 67-88.

Dates
First available in Project Euclid: 14 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1300108399

Digital Object Identifier
doi:10.14492/hokmj/1300108399

Mathematical Reviews number (MathSciNet)
MR2790830

Zentralblatt MATH identifier
1221.35283

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D05: Navier-Stokes equations [See also 35Q30]
Secondary: 42B35: Function spaces arising in harmonic analysis 42B30: $H^p$-spaces

Keywords
Navier-Stokes equation uniqueness Campanato space Hardy space

Citation

NAKAI, Eiichi; YONEDA, Tsuyoshi. Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations. Hokkaido Math. J. 40 (2011), no. 1, 67--88. doi:10.14492/hokmj/1300108399. https://projecteuclid.org/euclid.hokmj/1300108399


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